Solving Absolute Value Equations on Both Sides of the Equation
This page gives exercises and examples on solving absolute value equations on both sides of the equation. The exercises are in the next section.
In case you want to see the examples on solving absolute value equations on both sides first, go to the middle of the post for the examples and explanations.
Absolute Value – Exercises
1) If | x + y | = 5 and x = 3, then what is a possible value of y ?
2) What is the value of | xy | when x = –10 and y = 14?
3) If x = 8 and the value of | x ÷ y | = 0, then what is a possible value of y?
4) If x = –7 and | x | = y, then what is a possible value of y?
5) If y = 8 and | x | = y, what is a possible value of x?
Absolute Value Problems – Answers
Answer 1
1) The correct answer is: y = 2 and y = –8
If | x + y | = 5 and x = 3, we can substitute 3 for the value of x to solve.
| x + y | = 5
| 3 + y | = 5
| 3 + 2 | = 5
y = 2
However, for absolute value problems, you also have to consider other possible values. If | x + y | = 5 then | x + y | = | –5 | We know this because the absolute value of –5 is 5. So again substitute for the value of x to solve.
| x + y | = | –5 |
| 3 + y | = | –5 |
| 3 + –8 | = | –5 |
So, the possible values of y are –8 and 2.
Answer 2
2) The correct answer is: 140
What is the value of | xy | when x = –10 and y = 14?
Substitute the values to solve. | xy | = | –10 × 14 | = | – 140 | = 140
Answer 3
3) The answer is: no real number
If x = 8 and the value of | x ÷ y | = 0, then what is a possible value of y?
This could be considered a trick question. It is impossible to divide 8 by any real number that would result in zero as the answer.
Answer 4
4) The answer is: y = 7
If x = –7 and | x | = y, then what is a possible value of y?
| x | = y
| –7 | = y
7 = y
Answer 5
5) The answer is: x = –8 or x = 8
If y = 8 and | x | = y, what is a possible value of x?
| x | = y
| x | = 8
x = –8 or x = 8
Solving Absolute Value Equations on Both Sides – Explanations
Absolute value equations look like the following:
| –3 + 5 – 8 | = ?
For absolute value problems, you will see numbers, variables, or terms between the symbols | |.
The symbol means that you have to simplify the expression and then give the result as a positive number or variable.
So, the results of absolute value equations will always be positive.
Absolute Value — Example
Let’s look at our example problem again: | –3 + 5 – 8 | = ?
To solve the equation, perform these steps.
Step 1:
Simplify the equation inside the absolute value symbols.
–3 + 5 – 8 =
5 – 8 – 3 =
5 – 11 =
–6
Step 2:
Express the result as a positive number.
| –6 | = 6
Solving Absolute Value Equations on Both Sides – Advanced Problems
You may also see advanced problems on solving absolute value equations on both sides on your exam.
These types of questions may include algebraic expressions or inequalities.
For example: If x = 2 and y = 3, then | –x + 3y | = ?
Simplify the equation by substituting the values.
| –x + 3y | =
| –2 + (3 × 3) | =
| –2 + 9 | =
| 7 | =
7
Solving Absolute Value Equations on Both Sides – Rules to Remember
Addition
The absolute value of the sum of two numbers will always be less than or equal to the sum of the individual absolute values.
The equation for the addition rule is:
| x + y | ≤ | x | + | y |
Example:
If x = –7 and y = 5
| x + y | ≤ | x | + | y |
| –7 + 5 | ≤ | –7 | + | 5 |
| –2 | ≤ 7 + 5
2 ≤ 12
Subtraction
If the absolute value of the difference between two numbers is zero, then the two numbers must be equal.
The equation for the subtraction rule is:
If | x – y | = 0, then x = y
Example:
If x = 3 and y = 3
| 3 – 3 | = 0
0 = 0
Multiplication
The absolute value of the product of two numbers will always be equal to the product of the individual absolute values.
The equation for the multiplication rule is:
| xy | = | x | × | y |
Example:
If x = 4 and y = –8
| xy | = | x | × | y |
| 4 × –8 | = | 4 | × | –8 |
| –32 | = 4 × 8
| –32 | = 32
Division
The absolute value of the quotient of two numbers will always be equal to the quotient of the individual absolute values.
The equation for the division rule is:
| x ÷ y | = | x | ÷ | y |
Example:
If x = 6 and y = –3
| x ÷ y | = | x | ÷ | y |
| 6 ÷ –3 | = | 6 | ÷ | –3 |
| –2 | = 6 ÷ 3
| –2 | = 2
Solving Absolute Value Inequalities
1) If the absolute value of x is less than or equal to y, then the value of x will always be between the values of negative y and y.
The equation for the first inequality rule is:
If, | x | ≤ y, then –y ≤ x ≤ y
Example:
If x = –9 and y = 12
If | x | ≤ y, then –y ≤ x ≤ y
If | –9 | ≤ 12, then –12 ≤ –9 ≤ 12
2) If the absolute value of x is greater than or equal to y, then the value of x will be less than or equal to the value of negative y or y will be less than or equal to x.
The equation for the second inequality rule is:
If | x | ≥ y, then x ≤ –y or y ≤ x
Example:
If x = 15 and y = –5
If | x | ≥ y, then y ≤ x
If | 15 | ≥ –5, then –5 ≤ 15
You may also like to look at our posts on math help and geometry.