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Solving Absolute Value Equations

As we know, the absolute value of a number refers to its distance from 0 on the number line. It is represented as |x|, which defines the magnitude of any integer "x". The absolute value of any value, whether positive or negative, real or complex, will be a non negative real number, regardless of which sign it has. The operation is represented by two vertical lines, |x|, which we read as the absolute value of x.

For example, 8 is the absolute value for both +8 and -8.

| -8 | = 8 and | +8 | = 8

The absolute value of a number, whether the number is positive or negative, is always a non-negative number. The absolute value of 0 is 0.

Examples of absolute values

| -9 | = 9

| 17 | = 17

| 0 | = 0

| -27 | = 27

| 7 - 4 | = | 3 | = 3

| 12 - 24 | = | - 12 | = 12

| - 4 × 4 | = | - 16 | = 16

Properties of absolute values

There are several properties of absolute values that hold true in all equations. They are as follows:

  • Non-negativity: | x | is always greater than or equal to 0
  • Multiplicative: | x × y | = | x | × | y |
  • Subadditivity: | x + y | is always less than or equal to | x | + | y |

Solving absolute value equations

Absolute value equations are equations that have one or more variables in the absolute value bars.

When solving absolute value equations, there are two cases to consider:

Case 1: The expression inside the absolute value bars is positive.

Case 2: The expression inside the absolute value bars is negative.

Example 1

Take the expression | 4 x + 2 | = 18 as an example.

For this to be true, either

4 x + 2 = 18 or 4 x + 2 = -18

You need to solve both equations to get the correct answers to the absolute value equation.

Case 1:

4 x + 2 = 18

4 x = 16

x = 16 4 = 4

Case 2:

4 x + 2 = -18

4 x = -20

x = - 20 4 = -5

So there are two solutions to the absolute value equation | 4 x + 2 | = 18 , x = 4 , x = - 5 .

Example 2

It's also possible for an absolute value equation to have only one solution.

For example, the single solution to | x + 3 | = 0 because are cases break down into

Case 1:

| x + 3 | = -0

Case 2:

| x + 3 | = +0

which both have the same solution, x = -3

Example 3

It is also possible for an absolute value equation to have no solutions.

| 5 x + 1 | = -6

Since the absolute value of any expression is positive, there is no value of x for which this is true.

Practice working with absolute value equations

a. Arrange the given numbers in ascending order.

| 18 | , - | -7 | , | 17 | , | 93 | , | - 68 | , | 6 | , | - 7 |

First, find the absolute value of each number:

18 , - 7 , 17 , 93 , 68 , 6 , 7

Then arrange the numbers in order from smallest to largest:

- 7 , 6 , 7 , 17 , 18 , 68 , 93

b. Solve the equation | x + 4 | = 1 .

First, write the two equations from the given equation in such a way that

x + 4 = 1

x + 4 = -1

Solve the above equations to get the possible values of x.

x + 4 = 1

x = 1 - 4

x = - 3

x + 4 = -1

x = -1 - 4

x = - 5

So the two possible values of x are -3 and -5.

c. Solve the equation | 2 x - 5 | = 9

First, write the two equations from the given equation in such a way that

2 x - 5 = 9

2 x - 5 = -9

Solve the above equations to get the possible values of x.

Case 1:

2 x - 5 = 9

2 x = 9 + 5

2 x = 14

x = 14 2

x = 7

Case 2:

2 x - 5 = - 9

2 x = - 9 + 5

2 x = - 4

2 x = - 4 2

x = - 2

So the two possible values of x are 7 and -2.

Topics related to the Solving Absolute Value Equations

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Absolute Value Functions

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Get help learning about solving absolute value equations

The concept of absolute value and working with absolute value equations can be confusing for many students. Even if the idea of absolute value is easy to understand, working with it in equations can be tricky. If your student is having any trouble working with absolute value equations, having them work with a professional tutor would be an excellent idea. When compared to students who did not engage in tutoring, those who did take part in tutoring sessions increased their test scores and grades. A tutor can support your student's in-class learning while providing the undivided attention of an expert in math at their level while meeting in a location without the same distractions that are part of working in a classroom.

Your student's tutor can give continuous, in-the-moment feedback as your student works on absolute value equations so that they don't develop bad habits. Another benefit of working with a tutor is that they can discover the way your student learns best and teach them how to use strategies that take advantage of that to work out all types of math problems. Tutors can take all the time your student needs to fully understand a topic. A tutor can also break down complex concepts and processes into small steps so your student understands how absolute value equations work. If your student can benefit from the help of a private tutor, contact the Educational Directors at Varsity Tutors today to learn more about how tutoring can help your child.

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