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Unit 2 Tutorials: Solving Linear Equations
and Inequalities
INSIDE UNIT 2
Operations with Expressions
The FOIL Method
Rationalizing the Denominator
Terms and Factors in Algebraic Expressions
Properties in Algebraic Expressions
Solving Equations
Solving single-step equations
Isolating Variables
Solving Multi-step Equations
Literal Equations
Substitution in multi-step linear equations
Writing Equivalent Equations
Inequalities and Absolute Value
Introduction to Inequalities
Set Notation and Interval Notation
Solve Linear Inequalities
Compound Inequalities
Absolute Value Equations
Absolute Value Inequalities
Arithmetic Sequences & College Algebra in Context
Introduction to Arithmetic Sequences
Summation Notation
Finding the Sum of an Arithmetic Sequence
Distance, Rate, and Time
Determine an Equation in Context
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Solving Problems involving Percents
Solving Mixture Problems using Weighted Average
Converting Unit Rates
by Sophia Tutorial
WHAT’S COVERED
ïƒŠ
1. Identity Properties
2. Inverse Properties
3. Commutative Property of Addition and Multiplication
4. Associative Properties of Addition and Multiplication
5. Distributive Property
1. Identity Properties
The identity property of addition states that when zero is added to any number, the value does not change.
Below is an example of this property using real numbers:
In other words, the identity property of addition tells us that adding zero does not change the value of a
number. Generally, we can express this as:
ïƒƒ
FORMULA
A similar property applies to multiplication. What quantity, when multiplied to a number, does not change its
value? When any number is multiplied by 1, the value does not change. Let’s take a look at a numerical
example of the identity property of multiplication:
The identity property of multiplication states that any number multiplied by 1 does not change in value.
Generally, we can express this as:
ïƒƒ
FORMULA
Identity Property of Multiplication
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2. Inverse Properties
The inverse property of addition states that any number and its opposite sum to zero. We can refer to the
opposite of a number as its additive inverse. A numerical example is illustrated below:
9 + ( -9 ) = 0
9 and â€“9 are opposites of each other. This means that their magnitudes are the same (9), but their signs are
different; one is a positive number, while the other is a negative number. The sum of a number and its
opposite is zero. We can write this generally as:
ïƒƒ
FORMULA
The inverse property of multiplication states that a number and its reciprocal multiply to 1. In the same way
that a number and its opposite are additive inverses, a number and its reciprocal are multiplicative inverses.
The reciprocal of a number can be found by creating a fraction, and flipping the numerator and denominator.
Below is a numerical example of the inverse property of multiplication:
9 and 1/9 are multiplicative inverses, or reciprocals, of one another. The inverse property of multiplication
dictates that the product of a number and its reciprocal is equal to 1. Here is our general rule:
ïƒƒ
FORMULA
Inverse Property of Multiplication
3. Commutative Property of Addition and
Multiplication
Addition and multiplication is commutative. This means that we can add in any order we wish, and we can
multiply in any order we wish. It is important to note that we cannot mix addition and multiplication. These are
separate properties, but they behave the same with both operations.
ïŒ˜
TERM TO KNOW
Commutative Property
A property of addition that allows terms to be added in any order; a property of multiplication that
allows factors to be multiplied in any order.
Let’s take a look at a numerical example of the commutative property of addition:
The expression 2 + 3 is the same as the expression 3 + 2, because addition is commutative. It does not matter
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in which order you add, the sum will be 5 in either case.
ïƒƒ
FORMULA
The same property applies to multiplication as well. It does not matter in which order you multiply, because
multiplication is commutative. A numerical example is illustrated below:
The expression 3 x 4 is the same as the expression 4 x 3, because multiplication is commutative. It does not
matter in which order you multiply, the product will be 12 in either case.
ïƒƒ
FORMULA
Commutative Property of Multiplication
4. Associative Properties of Addition and
Multiplication
The associative property allows us to group terms for addition and multiplication in any way we wish. As with
the commutative properties of addition and multiplication, the associative property applies to addition and
multiplication separately.
ïŒ˜
TERM TO KNOW
Associative Property
A property of addition that allows terms to be grouped in any order; a property of multiplication
that allows factors to be grouped in any order.
Here are some numerical examples of the associative property:
(3 + 4) + 6 = 3 + (4 + 6)
4 x (2 x 8) = (4 x 2) x 8
The associative property allows us to group addends or group factors in different ways. This is particularly
helpful in mental math, where we might easily recognize that 4 + 6 is 10. In such cases, regrouping can help us
recognize certain sums or products to make mental math easier.
ïƒƒ
FORMULA
ïƒƒ
FORMULA
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5. Distributive Property
The distributive property applies multiplication over addition. This property is applied when we have a factor
being multiplied by a sum. It got its name from the process of distributing the outside factor into each added of
the sum.
ïŒ˜
TERM TO KNOW
Distributive Property
A property of multiplication that states that a sum multiplied by a factor can be expressed as a
sum of the products of each individual addend and that factor.
Let’s take a look at a numerical example:
2 (4 + 3)
Distribute 2 into 4 and 3
(2 x 4) + (2 x 3)
Multiply inside the parentheses
8+6
14
Our solution
ïƒƒ
FORMULA
Distributive Property
ï€¤
HINT
The distributive property is especially useful when working with variables. Realistically, an easier
approach with numerical examples is to evaluate what is inside the parentheses first, and then multiply the
outside factor. (This also follows the Order of Operations). However, as we work with algebraic
expressions containing variables, the distributive property is going to be very helpful.
ï‘¬
SUMMARY
There are many different properties for addition and multiplication. We looked at an identity property,
an inverse property, a commutative property, the associative property. Also, for multiplication, there
is the distributive property.
ï…œ
TERMS TO KNOW
Associative Property
A property of addition that allows terms to be grouped in any order; a property of multiplication that
allows factors to be grouped in any order.
Commutative Property
A property of addition that allows terms to be added in any order; a property of multiplication that allows
factors to be multiplied in any order.
Distributive Property
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A property of multiplication that states that a sum multiplied by a factor can be expressed as a sum of
the products of each original addend and that factor.
ïƒƒ
FORMULAS TO KNOW
Associative Property of Multiplication
Commutative Property of Multiplication
Distributive Property
Identity Property of Multiplication
Inverse Property of Multiplication
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The FOIL Method
by Sophia Tutorial
WHAT’S COVERED
ïƒŠ
1. Distributive Rule
2. The FOIL Method
3. Practice using FOIL
1. The Distributive Rule
Before we introduce the FOIL method, it is helpful to review the distributive rule, because the two are similar
processes. The distributive rule is used when a quantity is being multiplied by a sum. For example, 3(5 + 9)
can be evaluated using distribution. The 3 outside of (5 + 9) is multiplied into each term, written equivalently
as (3â€¢5)+(3â€¢9). We can evaluate this as the sum of 15 and 27, or 42. We can confirm this by evaluating 3(5 + 9)
using the order of operations: 3(5 + 9) = 3(14) = 42.
2. The FOIL Method
Earlier, we saw how distribution can help us evaluate expressions in the form a(b + c), by distributing a into b +
c to get ab + ac. How could we distribute something like (4 + 3)(5 + 1)?
We can evaluate such expressions by distributing, but the process works in a slightly different way. What really
happens is that we distribute twice: first, we distribute 4 into 5 and 1, then we distribute 3 into 5 and 1. Take a
look at how this distribution works:
Distribute 4 into
:
Distribute 3 into
:
Sum all parts
Our Solution
Taking a look at the distributions, we can say that in our first step, we multiplied the first terms of each factor. 4
is the first term in (4 + 3) and 5 is the first term in (5 + 1). Our next step was to multiply the two outer terms. 4
and 1 are the outermost terms in our expression. In the next step, we multiplied the two inner terms: 3 and 5.
And finally, we multiplied the last terms in each factor: 3 is the last term in (4 + 3) and 1 is the last term in (5 + 1).
In short, we multiplied the first term in each factor, then the outside two terms, then the inside two terms, and
then the last term in each factor. This is known as the FOIL method: First, Outside, Inside, Last.
ï€¤
HINT
Using FOIL to evaluate numerical expressions may seem odd, but FOIL is an extremely useful method
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when working with quadratics and algebraic expressions. Practicing without variables will help us see the
properties and relationship to distribution, which will make FOILing algebraic expressions much easier!
ïŒ˜
TERM TO KNOW
FOIL
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside,
inside, last.
3. Practice Using FOIL
Let’s take a look at some more examples of using FOIL to evaluate expressions. As we work through these
examples, pay attention to the sign of the numbers. We bring positive and negatives with us when distributing!
First:
Inside:
, Outside:
, Last:
Sum all parts
Our Solution
First:
Inside:
, Outside:
, Last:
Sum all parts
Our Solution
ï‘¬
SUMMARY
The distributive rule is used for the FOIL method when you’re multiplying groups of terms in the form
(a+b)(c+d). Remember, these are called binomials. If we’re multiplying (a+b) times (c+d), we are
multiplying two binomials. When we practice using FOIL, the acronym can help us remember the
steps for doing that distributing. It is important to remember that FOIL stands for First, Outer, Inner,
and Last.
ï…œ
TERMS TO KNOW
FOIL
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside,
last.
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ïƒƒ
FORMULAS TO KNOW
FOIL Method
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by Sophia Tutorial
WHAT’S COVERED
ïƒŠ
Expressions
example:
Combine like terms
Our Solution
Combine like terms
Our Solution
Notice that when we combined the terms with
it was just like combining terms with x. When adding and
subtracting with radicals we can combine like radicals just as like terms. We add and subtract the coefficients
in front of the radical, and the radical stays the same. This is shown in the following example.
and
Our Solution
We cannot simplify this expression any more as the radicals do not match. Often problems we solved have no
like radicals, however, if we simplify the radicals first we may find we do in fact have like radicals.
Simplify radicals, find perfect square factors.
Take roots where possible
Combine like terms
Our Solution
ï™
DID YOU KNOW
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The Arab writers of the 16th century used the symbol similar to the greater than symbol with a dot
This exact process can be used to add and subtract radicals with higher indices. (“Indices” is the plural of
index, which indicates the type of root).
Simplify each radical, finding perfect cube factors.
Take roots where possible
Multiply coefficients
Combine like terms
Our Solution
ï‘¬
SUMMARY
When adding and subtracting radical expressions, you can combine them if they are like terms.
Radicals are like terms if they have the same radicand and the same index. They have to have the
same number underneath the radical sign, and they have to have the same index, meaning they’re
both a square root or a cubed root or a fifth root. Sometimes you can break down a radicand into
factors to simplify the radicand, in which case you might be then able to combine it with another like
term.
Source: Adapted from “Beginning and Intermediate Algebra” by Tyler Wallace, an open source textbook
available at: http://wallace.ccfaculty.org/book/book.html
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by Sophia Tutorial
ïƒŠ
WHAT’S COVERED
1. Review of Distributive Rule and FOIL with Integers
2. Multiplying Radical Expressions using Distribution and FOIL
1. Review of Distributive Rule and FOIL
with Integers
The distributive rule allows us to distribute an outside factor into all terms of another factor. For example:
Distribute 2 into 4 and 3
Multiply inside the parentheses
Our Solution
If we have to factors in the form (a+b), we can use the distributive property in a different way, commonly
referred to as the FOIL method.
Apply steps to FOIL
Our Solution
Distribution and FOIL
The distributive rule and FOIL method can be applied to multiply expressions with radicals as well. First, we
will look at an example of distribution, where two identical radicals are multiplied together.
Distribute the square root of 2
simplifies to the integer 2
Our Solution
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We can also use the FOIL method to distribute across two binomials in multiplication when there are radicals.
This is illustrated in the following example:
Apply the steps to FOIL
simplifies to the integer 3
Combine like terms
Our Solution
ï€…
BIG IDEA
The distributive rule and the FOIL method can be applied to expressions containing radicals as well.
When two identical square roots are multiplied by each other, it evaluates to the expression underneath
the square root. This property also applies to other roots, such as cube roots, but the identical radical
needs to be multiplied by itself 3 times, and so on.
ïŒ˜
TERM TO KNOW
FOIL
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside,
inside, last.
ï‘¬
SUMMARY
A review of the distributive property and FOIL allows us to distribute outside factors into all terms of
another factor. A helpful hint when multiplying radical expressions using distribution and FOIL is to
remember that multiplying 2 square root terms together will cancel out the square root operation. For
example, the square root of 8 times the square root of 8 is just 8.
ï…œ
TERMS TO KNOW
FOIL
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside,
last.
ïƒƒ
FORMULAS TO KNOW
FOIL Method
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Rationalizing the Denominator
by Sophia Tutorial
WHAT’S COVERED
ïƒŠ
1. Rationalizing the Denominator
1. Rationalizing the Denominator
It is considered bad practice to have a radical in the denominator of a fraction. When this happens we multiply
the numerator and denominator by the same thing in order to clear the radical. In the lesson on dividing
radicals we talked about how this was done with monomials. Here we will look at how this is done with
binomials.
If the binomial is in the numerator the process to rationalize the denominator is essentially the same as with
monomials The only difference is we will have to distribute in the numerator.
ï€¤
HINT
It is important to remember that when reducing the fraction we cannot reduce with just the 3 and 12 or just
the 9 and 12. When we have addition or subtraction in the numerator or denominator we must divide all
terms by the same number. As we are rationalizing it will always be important to constantly check our
problem to see if it can be simplified more. We ask ourselves, can the fraction be reduced? Can the
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radicals be simplified? These steps may happen several times on our way to the solution.
If the binomial occurs in the denominator we will have to use a different strategy to clear the radical. Consider
, if we were to multiply the denominator by
with
we would have to distribute it and we would end up
. We have not cleared the radical, only moved it to another part of the denominator. So our
current method will not work. Instead we will use what is called a conjugate. A conjugate is made up of the
same terms, with the opposite sign in the middle. So for our example with
in the denominator, the
conjugate would be
ïŒ˜
TERM TO KNOW
Conjugate
The conjugate of a binomial is a binomial with the opposite sign between its terms.
The advantage of a conjugate is when we multiply them together we have
, which is a
difference and a sum. We know when we multiply these, we get a difference of squares. Squaring
and 5,
with subtraction in the middle gives the product:
Our answer when multiplying conjugates will no longer have a square root, which is exactly what we want.
ï€…
BIG IDEA
To rationalize a denominator containing a radical expression, multiply the fraction using its conjugate. The
product will no longer contain a radical.
ï¤ EXAMPLE
In the previous example, we could have reduced by dividing by -2 instead of 2, giving
are correct.
ï¤ EXAMPLE
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The same process can be used when there is a binomial in the numerator and denominator. We just need to
remember to FOIL out the numerator.
ï¤ EXAMPLE
ï¤ EXAMPLE
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ï™
DID YOU KNOW
During the 5th century BC in India, Aryabhata published a treatise on astronomy. His work included a
method for finding the square root of numbers that have many digits.
ï‘¬
SUMMARY
Rationalizing the denominator involves multiplying by a conjugate in both the denominator and the
numerator of a fraction and then simplifying. The reason that we do that is because having an
irrational radical in the denominator of a fraction is not considered simplified. The conjugate of the âˆša
+ b is just âˆša – b. We just use the opposite sign in between the two terms. The conjugate of just a
Source: Adapted from “Beginning and Intermediate Algebra” by Tyler Wallace, an open source textbook
available at: http://wallace.ccfaculty.org/book/book.html
ï…œ
TERMS TO KNOW
Conjugate
The conjugate of a binomial is a binomial with the opposite sign between its terms.
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Terms and Factors in Algebraic Expressions
by Sophia Tutorial
WHAT’S COVERED
ïƒŠ
1. Terms in Algebraic Expressions
2. Types of Algebraic Expressions
3. Parts of an Algebraic Expression
4. Combining Like Terms
1. Terms and Factors in Algebraic
Expressions
The following is a single term:
Here we see the two separate quantities 3 and
multiplied by one another. We can also dissect the
expression a little further we get 3â€¢xâ€¢xâ€¢xâ€¢xâ€¢xâ€¢xâ€¢x. Notice how all numbers and variables are combined through
multiplication only. We say that the above example represents a single term.
ïŒ˜
TERM TO KNOW
Term
A collection of numbers, variables, and powers combined through multiplication.
2. Types of Algebraic Expressions
When dealing with algebraic expressions the number of different terms that are added to or subtracted from
one another give the expression a different name. Here we will look at different types of expressions based
on the number of unique terms they contain.
The simplest algebraic expression is just a number, such as 3. A single number is called a constant, or a term
that is not multiplied by a variable.
A single algebraic expressions with no other terms added to or being subtracted from it is called a monomial.
For example,
ï€¤
is a monomial.
HINT
A constant is a special type of monomial where there are no variables being multiplied to a number. For
example, 5 or 14 are constants.
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Typically when writing algebraic expressions we refer to them using their variable and the power the variable
is being raised to. For example,
would be called a second degree monomial because it the variable is
being raised to the second power.
More complex types of algebraic expression contains more than one monomial and are combined through
If we have two monomials combined with one another we call that expression a binomial. For example,
and
are binomials.
If we have more than two monomials combined with one another we have what is called a polynomial. For
example,
is a polynomial. We typically say that this expression is a second degree polynomial
because the highest power of any variable in the expression is 2. If we had the expression
, we would count the total number of powers in each monomial to determine the
power. In this example we would say that this is a 3rd degree polynomial because in the term
the
combined power of x and y add up to 3.
3. Parts of an Algebraic Expression
When working with algebraic expressions you should be familiar with the parts that make up the expression.
Here we will discuss and identify coefficients, variables, powers or degree, and constant terms.
If we have the expression
, the variable represents an unknown quantity, and is typically written as
a letter. In this case the variable would be x. Coefficients would be the number in front of a variables. In this
case the 5 and 7 would be coefficients. The power or degree of this polynomial is 2 since that is the highest
power variables are being raised to. Finally, the constant would be 3 since that is the only term without a
variable component.
Now letâ€™s look at how to combine two or more algebraic expressions.
4. Combining Like-Terms
One way we can simplify expressions is to combine like-terms. Like-terms are terms where the variables
match exactly (exponents included). Examples of like-terms would be 3xy and âˆ’7xy or
and
or âˆ’3
and 5. If we have like-terms we are allowed to add (or subtract) the numbers in front of the variables, then
keep the variables the same. This is shown in the following examples.
Combine like terms
Combine like terms
Combine like terms
Our Solution
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Combine like terms
Combine like terms
Our Solution
As we combine like-terms, we need to interpret subtraction signs as part of the following term. This means if
we see a subtraction sign, we treat the following term like a negative term, the sign always stays with the term.
ï‘¬
SUMMARY
When considering terms and factors in algebraic expressions, we can define term as a collection of
numbers, variables, and powers. Types of algebraic expressions can be monomials, binomials, and
polynomials. Parts of an algebraic expression include the variables, corresponding coefficients,
powers, and constants. Terms are referred to by their variable and their power or exponent. When
combining like-terms, we are combining terms that have the same variable and the same power with
Source: Adapted from “Beginning and Intermediate Algebra” by Tyler Wallace, an open source textbook
available at: http://wallace.ccfaculty.org/book/book.html
ï…œ
TERMS TO KNOW
Algebraic Expression
A combination of numbers, variables, and operators representing a quantity.
Coefficient
The number in front of a variable term that acts as a factor or multiplier.
Constant
A term with no variable component.
Factor
A number or quantity used in multiplication.
Polynomial
An expression containing several terms.
Term
A collection of numbers, variables, and powers combined through multiplication.
Variable
A quantity that can change, expressed as a letter or symbol.
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Properties in Algebraic Expressions
by Sophia Tutorial
ïƒŠ
WHAT’S COVERED
1. Commutative Properties of Addition and Multiplication
2. Associative Properties of Addition and Multiplication
3. Distributive Property
and Multiplication
In short, the commutative properties of addition and multiplication allow us to add algebraic terms in any
order we wish, as well as multiply algebraic terms in any order we wish. These properties are illustrated in the
following examples:
To simplify, add 4x to 3x
Sum of 3x and 4x
To simplify, add 3x to 4x
Sum of 3x and 4x
To simplify, multiply 5 by 2c
Product of 5 and 2c
To simplify, multiply 2c by 5
Product of 2c and 5
In each example above, notice that the order in which applied the operation (either addition or multiplication)
did not affect the solution. It is important to note that subtraction and division are not commutative.
ï€…
BIG IDEA
Addition and multiplication is commutative. When adding terms, you can add them in any order you wish.
When multiplying terms, you may multiply the terms in any order you wish.
2. Associative Properties of Addition and
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Multiplication
The associative property deals with how terms of an expression are grouped together. For algebraic
expressions in which several terms are being added together, you can group terms together in any way in
order to make simplification easier. Here is an example of the associative property of addition:
We can group terms in any way
Our Solution
In some cases, such as the example above, the associative property is helpful when grouping like terms
together. We used the associative property first add 2a and 5a to get 7a, then we added 3 at the end.
The associative property holds true for multiplication as well, and works in a similar way to addition.
We can group terms in any way
Multiply x and 3x
Multiply by 2
Our Solution
3. Distributive Property
Often as we work with problems, there will be a set of parentheses that make solving a problem difficult, if not
impossible. To get rid of these unwanted parentheses, we can use the distributive property. Using this
property, we multiply the number in front of the parentheses by each term inside. Here are some examples:
Multiply each term by 4
Our Solution
Multiply each term by
Our Solution
ï€¤
HINT
Notice that in the previous example, we multiplied each term inside the parentheses by a negative
number. With the subtraction inside, this means we multiplied -7 by -6, to result in positive 42. The most
common error in distributing is a sign error. Be careful with your signs!
ï‘¬
SUMMARY
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The commutative, associative, distributive properties, as well as factoring, can be extended to
expressions that are involving variables. These properties will be useful when simplifying expressions
and solving equations.
Source: Adapted from “Beginning and Intermediate Algebra” by Tyler Wallace, an open source textbook
available at: http://wallace.ccfaculty.org/book/book.html
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Solving single-step equations
by Sophia Tutorial
ïƒŠ
WHAT’S COVERED
1. Solving single-step equations
b. Subtraction Problems
c. Multiplication Problems
d. Division Problems
1. Solving single-step equations
Solving linear equations is an important and fundamental skill in algebra. In algebra, we are often presented
with a problem where the answer is known, but part of the problem is missing. The missing part of the
problem is what we seek to find. An example of such a problem is shown below.
Notice the above problem has a missing part, or unknown, that is marked by x. If we are given that the
solution to this equation is âˆ’5, it could be plugged into the equation, replacing the x with âˆ’5. This is shown in
below:
Multiply
True!
Now the equation comes out to a true statement! Notice also that if another number, for example, 3, was
plugged in, we would not get a true statement.
Multiply
False!
Due to the fact that this is not a true statement, this demonstrates that 3 is not the solution. However,
depending on the complexity of the problem, this â€œguess and checkâ€ method is not very efficient. Thus, we
take a more algebraic approach to solving equations. Here we will focus on what are called â€œone-step
equationsâ€ or equations that only require one step to solve. While these equations often seem very
fundamental, it is important to master the pattern for solving these problems so we can solve more complex
problems.
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To solve equations, the general rule is to do the opposite. For example, consider the following example.
The 7 is added to the x
Subtract 7 from both sides to get rid of it
Our Solution
Then we get our solution, x = âˆ’ 12. The same process is used in each of the following examples.
1b. Subtraction Problems
In a subtraction problem, we get rid of negative numbers by adding them to both sides of the equation. For
example, consider the following example.
The 5 is negative, or subtracted from z
Our Solution
Then we get our solution x = 9. The same process is used in each of the following examples. Notice that each
time we are getting rid of a negative number by adding.
1c. Multiplication Problems
With a multiplication problem, we get rid of the number by dividing on both sides. For example consider the
following example.
Variable is multiplied by 4
Divide both sides by 4
Our Solution
Then we get our solution x=5
With multiplication problems it is very important that care is taken with signs. If x is multiplied by a negative
then we will divide by a negative. This is shown in the next example:
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Variable is multiplied by
Divide both sides by
Our Solution
The same process is used in each of the following examples. Notice how negative and positive numbers are
handled as each problem is solved.
1d. Division Problems
In division problems, we get rid of the denominator by multiplying on both sides. For example consider our
next example.
Variable is divided by
Multiply both sides by
Our Solution
Then we get our solution x = âˆ’ 15. The same process is used in each of the following examples.
The process described above is fundamental to solving equations. Once this process is mastered, the
problems we will see have several more steps. These problems may seem more complex, but the process and
patterns used will remain the same.
ï€…
BIG IDEA
To solve single-step equations, first identify the operation being applied to the variable. To isolate the
variable, apply the inverse operation. Addition and subtraction are inverses of each other; multiplication
and division are inverses of each other.
ï‘¬
SUMMARY
Solving single-step equations involves isolating the variable that you’re trying to solve for by using an
inverse operation. Any operation that you do on one side of the equation needs to be done on the
other side. You can always check your solution by substituting it in for the variable in your original
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equation and seeing if the statement holds true. This is good practice to get into when you’re doing
simple equations because when you do things more complicated, it’s more likely that you’re going to
be making a mistake.
Source: Adapted from “Beginning and Intermediate Algebra” by Tyler Wallace, an open source textbook
available at: http://wallace.ccfaculty.org/book/book.html
ï…œ
TERMS TO KNOW
Equation
a mathematical statement that two quantities or expressions are equal in value
Rule of Equality
any operation performed on one side of the equation must be performed on the other side, in order to
keep quantities equal in value
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Isolating Variables
by Sophia Tutorial
ïƒŠ
WHAT’S COVERED
1. Process of Solving an Equation
2. Review of Inverse Operations
3. Applying Inverse Operations
4. Simplifying Before Isolating a Variable
1. Process of Solving an Equation
When solving an equation for a variable, our main goal is to isolate a variable. In other words, we want to get
the variable by itself on one side of the equation, with all other expressions on the other side of the equals
sign. In this process, we must always remember that if we perform an operation on one side of the equal sing,
we must do the same on the other side of the equal sign. Let’s look at an example:
Solve for x
Subtract 2 from both sides
Divide both sides by 3
Our Solution
ï€…
BIG IDEA
Whatever we do on one side of the equation has to be done on the other side of the equation. This is
known as the Rule of Equality.
2. Review of Inverse Operations
When isolating a variable, we need to keep the following in mind:
Operation
Inverse Operation
Subtraction
Subtraction
Multiplication
Division
Division
Multiplication
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Powers
Roots
Roots
Powers
3. Applying Inverse Operations
A good rule of thumb is to isolate the outermost operations surrounding the variable first, working our way
inwards until we isolate the variable. Let’s look at an example:
Solve for x
Divide both sides by 2
Our Solution
ï€¤
HINT
In general, we apply the inverse operations following the reverse order of operations to isolate a variable.
4. Simplifying Before Isolating a Variable
Sometimes when we try to isolate a variable, it may be better to simplify the equation before we perform any
inverse operations. This is illustrated below:
Solve for x
There are two ways we can go about solving this equation. First, we can distribute 5 into the 2x and â€“6, and
then isolate the variable, or we can divide both sides of the equation by 5 first, and then solve for x. Either
method is value, and you are free to use either when trying to isolate the variable. Let’s take a look at how we
can use both methods to solve the equation above:
By distribution:
Solve for x
Distribute 5 into
Divide both sides by 10
Our Solution
Dividing 5 first:
Solve for x
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Divide both sides by 5
Express 7/5 as a decimal
Divide both sides by 2
Our Solution
Let’s look at another example where combining like terms before attempting to isolate the variable can be
Solve for x
Move the x terms to one side
Move constant term to one side
Divide both sides by 3
Our Solution
ï€¤
HINT
When trying to isolate a variable, it is always a good idea to simplify the equation as much as possible
before starting to isolate the variable with inverse operations. This usually means that we should combine
like terms whenever possible.
ï‘¬
SUMMARY
The process of solving an equation involves isolating the variable you want to solve for. When
isolating a variable, it is helpful to have a review of inverse operations: addition and subtraction are
inverse, multiplication and division are inverse and powers and roots are inverse. Keep in mind when
applying inverse operation that this will cancel the operations around the variable. Also, in using the
inverse operations, use the order of operations in reverse order. Finally, simplifying before isolating a
variable, such as distributing or combining like-terms, can be helpful.
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Solving Multi-step Equations
by Sophia Tutorial
ïƒŠ
WHAT’S COVERED
1. Solving an Equation
2. Multi-step Equations with Distribution
3. A Variable in the Denominator
4. Equations with a Root
1. Solving an Equation
When solving an equation, our main goal is to isolate the variable we wish to solve for onto one side of the
equation, and then evaluate the expression on the other side of the equation. To do this, we use inverse
operations to undo the operations on our variable. With single step equations, only one operation is applied
to our variable, so only one inverse operation must be applied. Let’s review these inverse operations.
Operation
Inverse Operation
Subtraction
Subtraction
Multiplication
Division
Division
Multiplication
Powers
Roots
Roots
Powers
What do we do when we have several operations being performed on the variable we wish to solve for? We
know that we must perform inverse operations to both sides of the equation (due to the Rule of Equality for
equations), but in what order? Think about the order of operations. These operations are applied to the
variable following the order of operations. To undo this, we apply inverse operations in the reverse order of
operations
ï€…
BIG IDEA
To solve multi-step equations, apply inverse operations to both sides of the equation, following the
reverse order of operations.
2. An Equation with Distribution
Let’s take a look at an example of an equation involving distribution. Generally, there are two options. Option 1
is to divide by the outside factor, eliminating the need to distribute anything. Option 2 is to simplify the
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expression by distributing the factor. We will see how to solve an equation both ways:
Divide the equation by 3
Undo x2 by dividing by 2
Our Solution
Distribute 3
Undo x6 by dividing by 6
Our Solution
3. A Variable in the Denominator
When a variable appears in the denominator of a fraction, it can be difficult to isolate that variable until it is
moved into a numerator. When the denominator contains the variable, the first step we take is to multiply the
entire equation by the expression in the denominator. This eliminates the variable from the denominator on
one side of the equation, and makes it part of the numerator on the other side of the equation. This is
illustrated in the example below:
Multiply the equation by 2x
Distribute 8 into 2x
Divide by 16
Our Solution
ï€¤
HINT
Notice that when clearing the variable in the denominator, we multiply the equation by the entire
denominator, not just by the variable alone.
4. An Equation involving a Power or a
Root
So far we have seen how to apply inverse operations to undo addition, subtraction, multiplication and division.
Lastly, let’s see how to apply power and roots to solve a multi-step equation.
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Undo the cube root by cubing both sides
Undo +3 by subtracting 3
Undo x12 by dividing 12
Our Solution
As we can see with the example above, we had an expression underneath a cube root. To undo this, we
cubed both sides of the equation. We were left with the expression alone, without a radical, and 3 cubed on
the other side. From there, we were able to solve for x by applying our familiar inverse operations.
ï‘¬
SUMMARY
The process for solving an equation involves using inverse operations in the reverse order of
operations or PEMDAS backwards. If possible, it is best to simplify the equation before using those
inverse operations. If a variable is in the denominator of a fraction, you need to multiply both sides of
the equation by that variable to solve the equation. Finally, if solving an equation with a power or
root, you need to isolate the radical and use a power to cancel it out before doing anything
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Literal Equations
by Sophia Tutorial
WHAT’S COVERED
ïƒŠ
1. Definition of Literal Equation
2. Formulas as Literal Equations
3. Rewriting Literal Equations
1. Definition of Literal Equation
A literal equation is an equation that has more than one variable. In math, we work with literal equations all the
time. For example, the slopeâ€“intercept form of a line is a literal equation: y = mx+b. This is because it has more
than one variable.
ïŒ˜
TERM TO KNOW
Literal equation
an equation with more than one variable
2. Formulas as Literal Equations
Formulas are common literal equations. Formulas relate variables together. For example, we can use a
formula to relate the length and width of a rectangle to its area. We can rewrite formulas to create expressions
for other variables in the equation. For example:
Formula for the area of a rectangle
Divide by w; expression for length
Divide by l, expression for width
3. Rewriting Literal Equations
We can rewrite literal equations to express other variables by apply inverse operations. More specifically, we
look at what operations are being applied to the variable we wish to isolate, as well as in what order they are
being applied. To isolate the variable, we apply the inverse operations in reverse order. This is shown below
with several common formulas:
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Formula for the area of a circle
Divide both sides by
Take square root of both sides
Formula for distance, rate, and time
Divide both sides by t
Formula for distance, rate, and time
Divide both sides by r
Pythagorean Theorem
Subtract
from both sides
Take square root of both sides
Pythagorean Theorem
Subtract
from both sides
Take square root of both sides
ï‘¬
SUMMARY
The definition of literal equations are equations that have more than one variable. Formulas are
literal equations and are used often in mathematics. Depending on what kind of information you are
given, you may wish to rewrite literal equations, or express the equations and formulas in different
ways.
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ï…œ
TERMS TO KNOW
Literal Equation
an equation with more than one variable
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Substitution in multi-step linear equations
by Sophia Tutorial
WHAT’S COVERED
ïƒŠ
1. Substitution Property of Equality
2. Substituting Expressions into Equations
3. Substituting to Solve an Equation
1. Substitution Property of Equality
The Substitution Property of Equality allows us to swap or substitute equivalent quantities in expressions and
equations. Let’s take a look at a basic example of substitution.
Substitute 3 in for x, because they are equal
Evaluate 2(3)
Our Solution
2. Substituting Expressions in Equations
Sometimes, we are given an expression for the variable, rather than a single value. We can still use the
Substitution Property of Equality to simplify expressions and solve equations. Most often, this requires
distribution after substituting, in order to simplify the equation or expression. This is illustrated in the example
below:
Substitute 3a-2 in for x, because they are equal
Distribute (.05) into (3a-2)
Combine like terms -1 and 12
Our Solution
ï€¤
HINT
If there is a coefficient in front of the variable that is substituted with an expression, it will require that we
distribute it into the newly substituted expression in order to simplify.
3. Substituting to Solve an Equation
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Page 37
Let’s apply the concept of substitution to solve an equation. Suppose you sell gift bags from a kiosk at a local
strip mall. Each gift bag costs \$7, and you received \$15 in tips for the day. We can represent your profit with
the equation:
, where R is revenue, and x is the number of gift bags sold. (7 is multiplied by x to
represent revenue from sales, and 15 is added to account for the tips.)
You figure that you averaged 8 sales per hour, and at the end of the day, a customer bought 10 of them for a
party she is attending. We can represent the number of gift bags sold by the equation
, where x is
the number of gift bags sold, and t is time in hours. (8 is multiplied by t to represent 8 bags sold each hour,
and we add 10 to account for the customer who bought 10 for her party.)
Let’s take a look at our equations:
How long did it take to generate \$253 in revenue? Notice that we can substitute \$253 in for R, but we want to
solve for t, time. One method would be to solve for x, and then substitute that value in order to solve for t.
Another method involves making all necessary algebraic substitutions first, and then solving a simplified
equation.
Substitute
in for x, because they are equal
Distribute 7 into
Combine like terms 70 and 15
Subtract 85 from both sides
Divide by 56
Our Solution
This means that \$253 was generated after 3 hours of selling gift bags. Not bad!
ï‘¬
SUMMARY
The substitution property of equality states that we can substitute an expression that is equal to
some variable into another equation or expression containing that same variable. After you substitute,
you can simplify and/or solve the equation.
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Writing Equivalent Equations
by Sophia Tutorial
ïƒŠ
WHAT’S COVERED
1. Equivalent Equations
2. Determining if Two Equations are Equivalent
1. Equivalent Equations
In mathematics, we work with equivalent equations all the time. Think about the process for solving a multistep equation. We might start with something such as 5x + 3 = 23 Using inverse operations, we create a series
of equivalent equations in order to find a value for x.
An equivalent equation
An equivalent equation
The equations above are all considered equivalent equations, because they have the same solution. In each
equation, the solution is x = 4.
2. Determining if Two Equations are
Equivalent
In order to determine if two equations are equivalent, we will solve each equation, and then compare their
solutions. If their solutions are the same, we can say the equations are equivalent. If the solutions are not the
same, we know that the equations are not equivalent.
Determine if the equations below are equivalent:
Solve each equation separately:
Subtract 6
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Subtract 9
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Divide by
Multiply by 2
The solutions to our equations are x = 2 and x = â€“2. Since the solutions are not the same, the two equations
are not equivalent.
ï‘¬
SUMMARY
We can define equivalent equations as equations that have the same solution or solution set. To
determine if two equations are equivalent to each other, you just need to solve each equation and
then determine if their solutions are the same.
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Introduction to Inequalities
by Sophia Tutorial
WHAT’S COVERED
ïƒŠ
1. What is an inequality?
2. Plotting inequalities on a number line
a. “Less Than” Inequalities
b. “Greater Than” Inequalities
c. Compound Inequalities
1. What is an inequality?
When we have an equation such as x = 4 we have a specific value for our variable. With inequalities we will
give a range of values for our variable. To do this we will not use equals, but one of the following symbols:
Greater than
Greater than or equal to
Less than
Less than or equal to
The above symbols are inequality symbols. They relate two quantities as not being equal to each other. Let’s
look at a definition of an inequality:
If we have an expression such as x < 4, this means our variable can be any number smaller than 4 such as âˆ’ 2, 0, 3, 3.9 or even 3.999999999 as long as it is smaller than 4. If we have an expression such as x greater or equal than âˆ’ 2, this means our variable can be any number greater than or equal to âˆ’2, such as 5, 0, âˆ’1, âˆ’1.9999, or even âˆ’2. ïŒ˜ TERM TO KNOW Inequality a mathematical statement that two quantities are not equal in value. 2. Plotting inequalities on a number line Because we donâ€™t have one set value for our variable, it is often useful to draw a picture of the solutions to the inequality on a number line. We will start from the value in the problem and bold the lower part of the number line if the variable is smaller than the number, and bold the upper part of the number line if the variable is larger. The value itself we will mark with brackets, either or for "less than" or "greater than" respectively, and or for "less than or equal to" or "greater than or equal to" respectively. Â© 2022 SOPHIA Learning, LLC. SOPHIA is a registered trademark of SOPHIA Learning, LLC. Page 41 Once the graph is drawn we can quickly convert the graph into what is called interval notation. Interval notation gives two numbers, the first is the smallest value, the second is the largest value. If there is no largest value, we can use âˆž (infinity). If there is no smallest value, we can use âˆ’ âˆž negative infinity. If we use either positive or negative infinity we will always use a curved bracket for that value. 2a. "Less Than" Inequalities Graph the inequality, x < 2, and give the interval notation. For the "less than" inequality, x < 2, we know that x cannot be exactly equal to two but can be anything smaller. First, we show that x cannot be exactly equal to two by putting a or an open circle on the number line right at the two. An open circle means that the value for x cannot be exactly equal to the value that it's on top of. We also want to show that x can be less than two. We can use an arrow pointing to the left, or pointing to the numbers that are less than two. The region above represents all the solutions to the inequality, x is less than 2, and any value in the highlighted region is going to satisfy this inequality. Keep in mind that if this inequality was x â‰¤ 2, then we would have used a bracket or a closed circle. 2b. "Greater Than" Inequalities Graph the inequality, y â‰¥ -1, and give the interval notation. For the "greater than or equal to" inequality, y â‰¥ -1, we need to show that y can be exactly equal to negative one. We can do that putting a or a closed circle on the number line right at negative one. Next, we want to show that y can also be anything greater than negative one. We can use an arrow pointing to the right, towards the numbers that are greater than negative one. This represents all the solutions to the inequality, y â‰¥ -1. Any value of y in this highlighted region, including negative one, is going to satisfy the inequality. 2c. Compound Inequalities Graph the inequality, , and give the internal notation. Â© 2022 SOPHIA Learning, LLC. SOPHIA is a registered trademark of SOPHIA Learning, LLC. Page 42 For the compound inequality, , the values for x that are going to satisfy this inequality have to be bigger than negative three and less than or equal to two. We start by putting a " "or an open circle at negative three because we see that x cannot be exactly equal to negative three. Then we are going to use the " " or closed circle at two because we know that x can be equal exactly to two. We want to show that x is greater than negative three but that x is also smaller than or equal to two. We can see that our solution is going to be in between these two points. Using this highlighted region, we can see that any value that's in this region is going to satisfy the inequality. Any number greater than negative three or less than or equal to two will satisfy this inequality. ï‘¬ SUMMARY When thinking about what is an inequality, we use inequalities to show that two quantities are not equal in value. We can use a number line to show a range of values that can satisfy an inequality expression. When we're plotting an inequality on a number line, we use an open circle or parentheses for the symbols "less-than" and "greater-than" and use a filled in circle or brackets for the symbols "less-than or equal to" and "greater-than or equal to." Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html ï…œ TERMS TO KNOW Inequality a mathematical statement that two quantities are not equal in value Â© 2022 SOPHIA Learning, LLC. SOPHIA is a registered trademark of SOPHIA Learning, LLC. Page 43 Set Notation and Interval Notation by Sophia Tutorial ïƒŠ WHAT'S COVERED 1. Set Notation 2. Interval Notation 3. Writing a Solution Set from a Number Line 1. Set Notation When looking at number lines with a range of highlighted values, we can write the range of values a couple of different ways: in a format called set notation, and in a format called interval notation. In set notation, we define the range of values as set of numbers, and we use curly braces to define the set, with a description of what is to be included in the set. Interval notation, on the other hand, describes an interval of values represented by the number line. Let's examine the number line below. We see that the highlighted values range from -2 to +3. Be sure to examine what kind of circles are used on the number line. Open circles indicate that the exact value of -2 is not included in the set, and the closed circle indicates that +3 is included in our set. This is going to have important implications on what kind of inequality symbols we use in set notation. In set notation, this is written as This is read as, "all x-values, such that x is between (but not including) -2, and 3 (including 3). ï€… BIG IDEA Set notation uses curly braces to define the number line solution as a set of values. Open circles corresponding to the strict inequalities < and >, while closed circles correspond to non-strict inequalities â‰¤
and â‰¥. In set notation, the vertical bar is read, “such that…”
2. Interval Notation
In interval notation, we define specific intervals on the number line, and enclose the range of values using
either square brackets
or parenthesis,
. Square brackets correspond to the closed circles on the number
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Page 44
line, or non-strict inequalities â‰¤ and â‰¥. The parentheses are used when there are open circles on the number
line, and correspond to the strict inequalities < and >. Also note that parentheses, never square brackets, are
always used with the infinities. This is because positive and negative infinity are not concretely defined as
numbers.
We can write the same number line as above using interval notation. Here is our number line with a
highlighted range of values.
As before, we see that the highlighted range is from -2 to +3. We do not include the exact value of -2, but we
do include the exact value of 3. So we will use parentheses to enclose -2, but square braces to enclose 3.
In interval notation, this is written as:
, or left parenthesis negative 2 comma 3 right square bracket. This
means that the solution on the number line is in the interval from (but not including) -2 up to, and including, 3.
ï€…
BIG IDEA
Interval notation describes an range of values from a starting point to a stopping point. We use either
square braces or parentheses, depending on if we are including or excluding exact values. Square braces
are used to include exact values, and parentheses
are used to exclude exact values. We always use
parentheses with the infinities.
3. Writing a Solution in Interval and Set
Notation
Let’s practice writing in set and interval notation, using number line solutions. Examine the number line below:
We notice that all values from 2 to positive infinity are highlighted. In set notation, we would write this as
. Note that our inequality symbol does not include the exact value of 2.
In interval notation, this is written as
. Note that we use a parenthesis to the left of 2, and also a
parenthesis to the right of infinity.
Let’s look at a more complicated example:
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This number line has two ranges highlighted. How do we write this in set notation and interval notation? We