how to solve absolute value function equations

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Number line.

• \left|3x+1\right|=4
• \left|a-6\right|=\left|6-a\right|
• 1.5\left|3p\right|=4.5
• 4\left|2x-8\right|=9
• \left|3x+4\right|=-3
• 5-\left|2(x+3)\right|=0

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How to solve absolute value equations

|x + 5| = 3.

Worksheet on Abs Val Equations Abs Val Eqn Solver

The General Steps to solve an absolute value equation are:

• Rewrite the absolute value equation as two separate equations, one positive and the other negative
• Solve each equation separately
• After solving, substitute your answers back into original equation to verify that you solutions are valid
• Write out the final solution or graph it as needed

It's always easiest to understand a math concept by looking at some examples so, check outthe many examples and practice problems below.

You can always check your work with our Absolute value equations solver too

Practice Problems

Example equation.

Solve the equation: | X + 5| = 3

Click here to practice more problems like this one , questions that involve variables on 1 side of the equation.

Some absolute value equations have variables both sides of the equation. However, that will not change the steps we're going to follow to solve the problem as the example below shows:

Solve the equation: |3 X | = X − 21

Solve the following absolute value equation: | 5X +20| = 80

Solve the following absolute value equation: | X | + 3 = 2 X

This first set of problems involves absolute values with x on just 1 side of the equation (like problem 2 ).

Solve the following absolute value equation: |3 X −6 | = 21

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• 1.6 Absolute Value Functions
• Introduction to Functions
• 1.1 Functions and Function Notation
• 1.2 Domain and Range
• 1.3 Rates of Change and Behavior of Graphs
• 1.4 Composition of Functions
• 1.5 Transformation of Functions
• 1.7 Inverse Functions
• Key Equations
• Key Concepts
• Review Exercises
• Practice Test
• Introduction to Linear Functions
• 2.1 Linear Functions
• 2.2 Graphs of Linear Functions
• 2.3 Modeling with Linear Functions
• 2.4 Fitting Linear Models to Data
• Introduction to Polynomial and Rational Functions
• 3.1 Complex Numbers
• 3.2 Quadratic Functions
• 3.3 Power Functions and Polynomial Functions
• 3.4 Graphs of Polynomial Functions
• 3.5 Dividing Polynomials
• 3.6 Zeros of Polynomial Functions
• 3.7 Rational Functions
• 3.8 Inverses and Radical Functions
• 3.9 Modeling Using Variation
• Introduction to Exponential and Logarithmic Functions
• 4.1 Exponential Functions
• 4.2 Graphs of Exponential Functions
• 4.3 Logarithmic Functions
• 4.4 Graphs of Logarithmic Functions
• 4.5 Logarithmic Properties
• 4.6 Exponential and Logarithmic Equations
• 4.7 Exponential and Logarithmic Models
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• Introduction to Trigonometric Functions
• 5.2 Unit Circle: Sine and Cosine Functions
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Learning Objectives

• Graph an absolute value function.
• Solve an absolute value equation.
• Solve an absolute value inequality.

Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away. Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years. Today, astronomers can detect galaxies that are billions of light years away. Distances in the universe can be measured in all directions. As such, it is useful to consider distance as an absolute value function. In this section, we will investigate absolute value functions .

Understanding Absolute Value

Recall that in its basic form f ( x ) = | x | , f ( x ) = | x | , the absolute value function, is one of our toolkit functions. The absolute value function is commonly thought of as providing the distance the number is from zero on a number line. Algebraically, for whatever the input value is, the output is the value without regard to sign.

Absolute Value Function

The absolute value function can be defined as a piecewise function

Determine a Number within a Prescribed Distance

Describe all values x x within or including a distance of 4 from the number 5.

We want the distance between x x and 5 to be less than or equal to 4. We can draw a number line, such as the one in Figure 2 , to represent the condition to be satisfied.

The distance from x x to 5 can be represented using the absolute value as | x − 5 | . | x − 5 | . We want the values of x x that satisfy the condition | x − 5 | ≤ 4. | x − 5 | ≤ 4.

So | x − 5 | ≤ 4 | x − 5 | ≤ 4 is equivalent to 1 ≤ x ≤ 9. 1 ≤ x ≤ 9.

However, mathematicians generally prefer absolute value notation.

Describe all values x x within a distance of 3 from the number 2.

Resistance of a Resistor

Electrical parts, such as resistors and capacitors, come with specified values of their operating parameters: resistance, capacitance, etc. However, due to imprecision in manufacturing, the actual values of these parameters vary somewhat from piece to piece, even when they are supposed to be the same. The best that manufacturers can do is to try to guarantee that the variations will stay within a specified range, often ±1%, ± 5%, ±1%, ± 5%, or ± 10% . ± 10% .

Suppose we have a resistor rated at 680 ohms, ± 5 % . ± 5 % . Use the absolute value function to express the range of possible values of the actual resistance.

5% of 680 ohms is 34 ohms. The absolute value of the difference between the actual and nominal resistance should not exceed the stated variability, so, with the resistance R R in ohms,

Students who score within 20 points of 80 will pass a test. Write this as a distance from 80 using absolute value notation.

Graphing an Absolute Value Function

The most significant feature of the absolute value graph is the corner point at which the graph changes direction. This point is shown at the origin in Figure 3 .

Figure 4 shows the graph of y = 2 | x – 3 | + 4. y = 2 | x – 3 | + 4. The graph of y = | x | y = | x | has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units. This means that the corner point is located at ( 3 , 4 ) ( 3 , 4 ) for this transformed function.

Writing an Equation for an Absolute Value Function

Write an equation for the function graphed in Figure 5 .

The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. See Figure 6 .

We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function. Instead, the width is equal to 1 times the vertical distance as shown in Figure 7 .

From this information we can write the equation

Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression. Note also that if the vertical stretch factor is negative, there is also a reflection about the x-axis.

If we couldn’t observe the stretch of the function from the graphs, could we algebraically determine it?

Yes. If we are unable to determine the stretch based on the width of the graph, we can solve for the stretch factor by putting in a known pair of values for x x and f ( x ) . f ( x ) .

Now substituting in the point (1, 2)

Write the equation for the absolute value function that is horizontally shifted left 2 units, is vertically reflected, and vertically shifted up 3 units.

Do the graphs of absolute value functions always intersect the vertical axis? The horizontal axis?

Yes, they always intersect the vertical axis. The graph of an absolute value function will intersect the vertical axis when the input is zero.

No, they do not always intersect the horizontal axis. The graph may or may not intersect the horizontal axis, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 8 ).

Solving an Absolute Value Equation

Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as 8 = | 2 x − 6 | , 8 = | 2 x − 6 | , we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

Solutions to Absolute Value Equations

For real numbers A A and B , B , an equation of the form | A | = B , | A | = B , with B ≥ 0 , B ≥ 0 , will have solutions when A = B A = B or A = − B . A = − B . If B < 0 , B < 0 , the equation | A | = B | A | = B has no solution.

Given the formula for an absolute value function, find the horizontal intercepts of its graph .

• Isolate the absolute value term.
• Use | A | = B | A | = B to write A = B A = B or −A = B , −A = B , assuming B > 0. B > 0.
• Solve for x . x .

Finding the Zeros of an Absolute Value Function

For the function f ( x ) = | 4 x + 1 | − 7 f ( x ) = | 4 x + 1 | − 7 , find the values of x x such that f ( x ) = 0 f ( x ) = 0 .

The function outputs 0 when x = 1.5 x = 1.5 or x = − 2. x = − 2. See Figure 9 .

For the function f ( x ) = | 2 x − 1 | − 3 , f ( x ) = | 2 x − 1 | − 3 , find the values of x x such that f ( x ) = 0. f ( x ) = 0.

Should we always expect two answers when solving | A | = B ? | A | = B ?

No. We may find one, two, or even no answers. For example, there is no solution to 2 + | 3 x − 5 | = 1. 2 + | 3 x − 5 | = 1.

Given an absolute value equation, solve it.

• Use | A | = B | A | = B to write A = B A = B or A = −B . A = −B .

Solve 1 = 4 | x − 2 | + 2. 1 = 4 | x − 2 | + 2.

Isolating the absolute value on one side of the equation gives the following.

The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.

In Example 5 , if f ( x ) = 1 f ( x ) = 1 and g ( x ) = 4 | x − 2 | + 2 g ( x ) = 4 | x − 2 | + 2 were graphed on the same set of axes, would the graphs intersect?

No. The graphs of f f and g g would not intersect, as shown in Figure 10 . This confirms, graphically, that the equation 1 = 4 | x − 2 | + 2 1 = 4 | x − 2 | + 2 has no solution.

Find where the graph of the function f ( x ) = − | x + 2 | + 3 f ( x ) = − | x + 2 | + 3 intersects the horizontal and vertical axes.

Solving an Absolute Value Inequality

Absolute value equations may not always involve equalities. Instead, we may need to solve an equation within a range of values. We would use an absolute value inequality to solve such an equation. An absolute value inequality is an equation of the form

where an expression A A (and possibly but not usually B B ) depends on a variable x . x . Solving the inequality means finding the set of all x x that satisfy the inequality. Usually this set will be an interval or the union of two intervals.

There are two basic approaches to solving absolute value inequalities: graphical and algebraic. The advantage of the graphical approach is we can read the solution by interpreting the graphs of two functions. The advantage of the algebraic approach is it yields solutions that may be difficult to read from the graph.

For example, we know that all numbers within 200 units of 0 may be expressed as

Suppose we want to know all possible returns on an investment if we could earn some amount of money within $200 of $600. We can solve algebraically for the set of values x x such that the distance between x x and 600 is less than 200. We represent the distance between x x and 600 as | x − 600 | . | x − 600 | .

This means our returns would be between $400 and $800.

Sometimes an absolute value inequality problem will be presented to us in terms of a shifted and/or stretched or compressed absolute value function, where we must determine for which values of the input the function’s output will be negative or positive.

Given an absolute value inequality of the form | x − A | ≤ B | x − A | ≤ B for real numbers a a and b b where b b is positive, solve the absolute value inequality algebraically.

• Find boundary points by solving | x − A | = B . | x − A | = B .
• Test intervals created by the boundary points to determine where | x − A | ≤ B . | x − A | ≤ B .
• Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation.

Solve | x − 5 | < 4. | x − 5 | < 4.

With both approaches, we will need to know first where the corresponding equality is true. In this case we first will find where | x − 5 | = 4. | x − 5 | = 4. We do this because the absolute value is a function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4. Solve | x − 5 | = 4. | x − 5 | = 4.

After determining that the absolute value is equal to 4 at x = 1 x = 1 and x = 9 , x = 9 , we know the graph can change only from being less than 4 to greater than 4 at these values. This divides the number line up into three intervals:

To determine when the function is less than 4, we could choose a value in each interval and see if the output is less than or greater than 4, as shown in Table 1 .

Because 1 < x < 9 1 < x < 9 is the only interval in which the output at the test value is less than 4, we can conclude that the solution to | x − 5 | < 4 | x − 5 | < 4 is 1 < x < 9 , 1 < x < 9 , or ( 1 , 9 ) . ( 1 , 9 ) .

To use a graph, we can sketch the function f ( x ) = | x − 5 | . f ( x ) = | x − 5 | . To help us see where the outputs are 4, the line g ( x ) = 4 g ( x ) = 4 could also be sketched as in Figure 11 .

We can see the following:

• The output values of the absolute value are equal to 4 at x = 1 x = 1 and x = 9. x = 9.
• The graph of f f is below the graph of g g on 1 < x < 9. 1 < x < 9. This means the output values of f ( x ) f ( x ) are less than the output values of g ( x ) . g ( x ) .
• The absolute value is less than or equal to 4 between these two points, when 1 < x < 9. 1 < x < 9. In interval notation, this would be the interval ( 1 , 9 ) . ( 1 , 9 ) .

For absolute value inequalities,

The < < or > > symbol may be replaced by ≤  or  ≥ . ≤  or  ≥ .

So, for this example, we could use this alternative approach.

Solve | x + 2 | ≤ 6. | x + 2 | ≤ 6.

Given an absolute value function, solve for the set of inputs where the output is positive (or negative).

• Set the function equal to zero, and solve for the boundary points of the solution set.
• Use test points or a graph to determine where the function’s output is positive or negative.

Using a Graphical Approach to Solve Absolute Value Inequalities

Given the function f ( x ) = − 1 2 | 4 x − 5 | + 3 , f ( x ) = − 1 2 | 4 x − 5 | + 3 , determine the x - x - values for which the function values are negative.

We are trying to determine where f ( x ) < 0 , f ( x ) < 0 , which is when − 1 2 | 4 x − 5 | + 3 < 0. − 1 2 | 4 x − 5 | + 3 < 0. We begin by isolating the absolute value.

Next we solve for the equality | 4 x − 5 | = 6. | 4 x − 5 | = 6.

Now, we can examine the graph of f f to observe where the output is negative. We will observe where the branches are below the x -axis. Notice that it is not even important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at x = − 1 4 x = − 1 4 and x = 11 4 x = 11 4 and that the graph has been reflected vertically. See Figure 12 .

We observe that the graph of the function is below the x -axis left of x = − 1 4 x = − 1 4 and right of x = 11 4 . x = 11 4 . This means the function values are negative to the left of the first horizontal intercept at x = − 1 4 , x = − 1 4 , and negative to the right of the second intercept at x = 11 4 . x = 11 4 . This gives us the solution to the inequality.

In interval notation, this would be ( − ∞ , − 0.25 ) ∪ ( 2.75 , ∞ ) . ( − ∞ , − 0.25 ) ∪ ( 2.75 , ∞ ) .

Solve − 2 | k − 4 | ≤ − 6. − 2 | k − 4 | ≤ − 6.

Access these online resources for additional instruction and practice with absolute value.

• Graphing Absolute Value Functions
• Graphing Absolute Value Functions 2
• Equations of Absolute Value Function
• Equations of Absolute Value Function 2
• Solving Absolute Value Equations

1.6 Section Exercises

How do you solve an absolute value equation?

How can you tell whether an absolute value function has two x -intercepts without graphing the function?

When solving an absolute value function, the isolated absolute value term is equal to a negative number. What does that tell you about the graph of the absolute value function?

How can you use the graph of an absolute value function to determine the x -values for which the function values are negative?

How do you solve an absolute value inequality algebraically?

Describe all numbers x x that are at a distance of 4 from the number 8. Express this using absolute value notation.

Describe all numbers x x that are at a distance of 1 2 1 2 from the number −4. Express this using absolute value notation.

Describe the situation in which the distance that point x x is from 10 is at least 15 units. Express this using absolute value notation.

Find all function values f ( x ) f ( x ) such that the distance from f ( x ) f ( x ) to the value 8 is less than 0.03 units. Express this using absolute value notation.

For the following exercises, solve the equations below and express the answer using set notation.

| x + 3 | = 9 | x + 3 | = 9

| 6 − x | = 5 | 6 − x | = 5

| 5 x − 2 | = 11 | 5 x − 2 | = 11

| 4 x − 2 | = 11 | 4 x − 2 | = 11

2 | 4 − x | = 7 2 | 4 − x | = 7

3 | 5 − x | = 5 3 | 5 − x | = 5

3 | x + 1 | − 4 = 5 3 | x + 1 | − 4 = 5

5 | x − 4 | − 7 = 2 5 | x − 4 | − 7 = 2

0 = − | x − 3 | + 2 0 = − | x − 3 | + 2

2 | x − 3 | + 1 = 2 2 | x − 3 | + 1 = 2

| 3 x − 2 | = 7 | 3 x − 2 | = 7

| 3 x − 2 | = − 7 | 3 x − 2 | = − 7

| 1 2 x − 5 | = 11 | 1 2 x − 5 | = 11

| 1 3 x + 5 | = 14 | 1 3 x + 5 | = 14

− | 1 3 x + 5 | + 14 = 0 − | 1 3 x + 5 | + 14 = 0

For the following exercises, find the x - and y -intercepts of the graphs of each function.

f ( x ) = 2 | x + 1 | − 10 f ( x ) = 2 | x + 1 | − 10

f ( x ) = 4 | x − 3 | + 4 f ( x ) = 4 | x − 3 | + 4

f ( x ) = − 3 | x − 2 | − 1 f ( x ) = − 3 | x − 2 | − 1

f ( x ) = − 2 | x + 1 | + 6 f ( x ) = − 2 | x + 1 | + 6

For the following exercises, solve each inequality and write the solution in interval notation.

| x − 2 | > 10 | x − 2 | > 10

2 | v − 7 | − 4 ≥ 42 2 | v − 7 | − 4 ≥ 42

| 3 x − 4 | ≤ 8 | 3 x − 4 | ≤ 8

| x − 4 | ≥ 8 | x − 4 | ≥ 8

| 3 x − 5 | ≥ 13 | 3 x − 5 | ≥ 13

| 3 x − 5 | ≥ − 13 | 3 x − 5 | ≥ − 13

| 3 4 x − 5 | ≥ 7 | 3 4 x − 5 | ≥ 7

| 3 4 x − 5 | + 1 ≤ 16 | 3 4 x − 5 | + 1 ≤ 16

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph.

y = | x − 1 | y = | x − 1 |

y = | x + 1 | y = | x + 1 |

y = | x | + 1 y = | x | + 1

For the following exercises, graph the given functions by hand.

y = | x | − 2 y = | x | − 2

y = − | x | y = − | x |

y = − | x | − 2 y = − | x | − 2

y = − | x − 3 | − 2 y = − | x − 3 | − 2

f ( x ) = − | x − 1 | − 2 f ( x ) = − | x − 1 | − 2

f ( x ) = − | x + 3 | + 4 f ( x ) = − | x + 3 | + 4

f ( x ) = 2 | x + 3 | + 1 f ( x ) = 2 | x + 3 | + 1

f ( x ) = 3 | x − 2 | + 3 f ( x ) = 3 | x − 2 | + 3

f ( x ) = | 2 x − 4 | − 3 f ( x ) = | 2 x − 4 | − 3

f ( x ) = | 3 x + 9 | + 2 f ( x ) = | 3 x + 9 | + 2

f ( x ) = − | x − 1 | − 3 f ( x ) = − | x − 1 | − 3

f ( x ) = − | x + 4 | − 3 f ( x ) = − | x + 4 | − 3

f ( x ) = 1 2 | x + 4 | − 3 f ( x ) = 1 2 | x + 4 | − 3

Use a graphing utility to graph f ( x ) = 10 | x − 2 | f ( x ) = 10 | x − 2 | on the viewing window [ 0 , 4 ] . [ 0 , 4 ] . Identify the corresponding range. Show the graph.

Use a graphing utility to graph f ( x ) = − 100 | x | + 100 f ( x ) = − 100 | x | + 100 on the viewing window [ − 5 , 5 ] . [ − 5 , 5 ] . Identify the corresponding range. Show the graph.

For the following exercises, graph each function using a graphing utility. Specify the viewing window.

f ( x ) = − 0.1 | 0.1 ( 0.2 − x ) | + 0.3 f ( x ) = − 0.1 | 0.1 ( 0.2 − x ) | + 0.3

f ( x ) = 4 × 10 9 | x − ( 5 × 10 9 ) | + 2 × 10 9 f ( x ) = 4 × 10 9 | x − ( 5 × 10 9 ) | + 2 × 10 9

For the following exercises, solve the inequality.

| − 2 x − 2 3 ( x + 1 ) | + 3 > −1 | − 2 x − 2 3 ( x + 1 ) | + 3 > −1

If possible, find all values of a a such that there are no x - x - intercepts for f ( x ) = 2 | x + 1 | + a . f ( x ) = 2 | x + 1 | + a .

If possible, find all values of a a such that there are no y y -intercepts for f ( x ) = 2 | x + 1 | + a . f ( x ) = 2 | x + 1 | + a .

Real-World Applications

Cities A and B are on the same east-west line. Assume that city A is located at the origin. If the distance from city A to city B is at least 100 miles and x x represents the distance from city B to city A, express this using absolute value notation.

The true proportion p p of people who give a favorable rating to Congress is 8% with a margin of error of 1.5%. Describe this statement using an absolute value equation.

Students who score within 18 points of the number 82 will pass a particular test. Write this statement using absolute value notation and use the variable x x for the score.

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using x x as the diameter of the bearing, write this statement using absolute value notation.

The tolerance for a ball bearing is 0.01. If the true diameter of the bearing is to be 2.0 inches and the measured value of the diameter is x x inches, express the tolerance using absolute value notation.

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• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: Precalculus 2e
• Publication date: Dec 21, 2021
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Absolute Value Functions

Solve an absolute value equation.

Now that we can graph an absolute value function, we will learn how to solve an absolute value equation. To solve an equation such as [latex]{8}=\left|{2}x - {6}\right|[/latex], we notice that the absolute value will be equal to 8 if the quantity inside the absolute value is 8 or -8. This leads to two different equations we can solve independently.

Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

An absolute value equation is an equation in which the unknown variable appears in absolute value bars. For example,

A General Note: Solutions to Absolute Value Equations

For real numbers [latex]A[/latex] and [latex]B[/latex], an equation of the form [latex]|A|=B[/latex], with [latex]B\ge 0[/latex], will have solutions when [latex]A=B[/latex] or [latex]A=-B[/latex]. If [latex]B<0[/latex], the equation [latex]|A|=B[/latex] has no solution.

How To: Given the formula for an absolute value function, find the horizontal intercepts of its graph.

• Isolate the absolute value term.
• Use [latex]|A|=B[/latex] to write [latex]A=B[/latex] or [latex]\mathrm{-A}=B[/latex], assuming [latex]B>0[/latex].
• Solve for [latex]x[/latex].

Example 4: Finding the Zeros of an Absolute Value Function

For the function [latex]f\left(x\right)=|4x+1|-7[/latex] , find the values of [latex]x[/latex] such that [latex]\text{ }f\left(x\right)=0[/latex] .

[latex]\begin{cases}0=|4x+1|-7\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Substitute 0 for }f\left(x\right).\hfill \\ 7=|4x+1|\hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \text{Isolate the absolute value on one side of the equation}.\hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ 7=4x+1\hfill & \text{or}\hfill & \hfill & \hfill & \hfill & -7=4x+1\hfill & \text{Break into two separate equations and solve}.\hfill \\ 6=4x\hfill & \hfill & \hfill & \hfill & \hfill & -8=4x\hfill & \hfill \\ \hfill & \hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ x=\frac{6}{4}=1.5\hfill & \hfill & \hfill & \hfill & \hfill & \text{ }x=\frac{-8}{4}=-2\hfill & \hfill \end{cases}[/latex]

The function outputs 0 when [latex]x=1.5[/latex] or [latex]x=-2[/latex].

For the function [latex]f\left(x\right)=|2x - 1|-3[/latex], find the values of [latex]x[/latex] such that [latex]f\left(x\right)=0[/latex].

Should we always expect two answers when solving [latex]|A|=B?[/latex]

No. We may find one, two, or even no answers. For example, there is no solution to  [latex]2+|3x - 5|=1[/latex].

How To: Given an absolute value equation, solve it.

• Use [latex]|A|=B[/latex] to write [latex]A=B[/latex] or [latex]A=\mathrm{-B}[/latex].

Example 5: Solving an Absolute Value Equation

Solve [latex]1=4|x - 2|+2[/latex].

Isolating the absolute value on one side of the equation gives the following.

The absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. At this point, we notice that this equation has no solutions.

In Example 5, if [latex]f\left(x\right)=1[/latex] and [latex]g\left(x\right)=4|x - 2|+2[/latex] were graphed on the same set of axes, would the graphs intersect?

No. The graphs of [latex]f[/latex] and [latex]g[/latex] would not intersect. This confirms, graphically, that the equation [latex]1=4|x - 2|+2[/latex] has no solution.

Find where the graph of the function [latex]f\left(x\right)=-|x+2|+3[/latex] intersects the horizontal and vertical axes.

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To solve an equation containing absolute value, isolate the absolute value on one side of the equation. Then set its contents equal to both the positive and negative value of the number on the other side of the equation and solve both equations.

Solve | x | + 2 = 5. 

Isolate the absolute value.

Set the contents of the absolute value portion equal to +3 and –3.

Answer: 3, –3 

Solve 3| x – 1| – 1 = 11. 

Set the contents of the absolute value portion equal to +4 and –4.

Solving for x,

Answer: 5, –3 

Solving inequalities containing absolute value and graphing

To solve an inequality containing absolute value , begin with the same steps as for solving equations with absolute value. When creating the comparisons to both the + and – of the other side of the inequality, reverse the direction of the inequality when comparing with the negative. 

Solve and graph the answer: | x – 1| > 2. 

Notice that the absolute value expression is already isolated.

| x – 1| > 2 

Compare the contents of the absolute value portion to both 2 and –2. Be sure to reverse the direction of the inequality when comparing it with –2.

Solve for x . 

Graph the answer (see Figure 1). 

Figure 1. The graphic solution to | x – 1| > 2.

Solve and graph the answer: 3| x | – 2 ≤ 1. 

Compare the contents of the absolute value portion to both 1 and –1. Be sure to reverse the direction of the inequality when comparing it with –1.

Graph the answer (see Figure 2). 

Figure 2. Graphing the solution to 3| x | – 2 ≤ 1.

Solve and graph the answer: 2|1 – x | + 1 ≥ 3. 

(Remember to switch the direction of the inequality when dividing by a negative)

Graph the answer (see Figure 3). 

Figure 3. Graphing the solution 2|1 – x | + 1 ≥ 3.

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Unit 10: Absolute value & piecewise functions

About this unit, graphs of absolute value functions.

• Shifting absolute value graphs (Opens a modal)
• Scaling & reflecting absolute value functions: equation (Opens a modal)
• Scaling & reflecting absolute value functions: graph (Opens a modal)
• Graphing absolute value functions (Opens a modal)
• Absolute value graphs review (Opens a modal)
• Shift absolute value graphs Get 3 of 4 questions to level up!
• Scale & reflect absolute value graphs Get 3 of 4 questions to level up!
• Graph absolute value functions Get 3 of 4 questions to level up!

Piecewise functions

• Introduction to piecewise functions (Opens a modal)
• Worked example: evaluating piecewise functions (Opens a modal)
• Worked example: graphing piecewise functions (Opens a modal)
• Worked example: domain & range of step function (Opens a modal)
• Worked example: domain & range of piecewise linear functions (Opens a modal)
• Absolute value & piecewise functions: FAQ (Opens a modal)
• Evaluate piecewise functions Get 3 of 4 questions to level up!
• Evaluate step functions Get 3 of 4 questions to level up!
• Piecewise functions graphs Get 3 of 4 questions to level up!

Absolute Value Function

An absolute value function is an important function in algebra that consists of the variable in the absolute value bars. The general form of the absolute value function is f(x) = a |x - h| + k and the most commonly used form of this function is f(x) = |x|, where a = 1 and h = k = 0. The range of this function f(x) = |x| is always non-negative and on expanding the absolute value function f(x) = |x|, we can write it as x, if x ≥ 0 and -x, if x < 0.

In this article, we will explore the definition, various properties, and formulas of the absolute value function. We will learn graphing absolute value functions and determine the horizontal and vertical shifts in their graph. We shall solve various examples based related to the function for a better understanding of the concept.

What is Absolute Value Function?

An absolute value function is a function in algebra where the variable is inside the absolute value bars. This function is also known as the modulus function and the most commonly used form of the absolute value function is f(x) = |x|, where x is a real number. Generally, we can represent the absolute value function as, f(x) = a |x - h| + k, where a represents how far the graph stretches vertically, h represents the horizontal shift and k represents the vertical shift from the graph of f(x) = |x|. If the value of 'a' is negative, the graph opens downwards and if it is positive, the graph opens upwards.

Absolute Value Function Definition

The absolute value function is defined as an algebraic expression in absolute bar symbols. Such functions are commonly used to find distance between two points . Some of the examples of absolute value functions are:

• g(x) = |3x - 7|
• f(x) = |-x + 9|

All the above given absolute value functions have non-negative values, that is, their range is all real numbers except negative numbers. All these functions change their nature ( increasing or decreasing ) after a point. We can find those points by expressing the absolute value function f(x) = a |x - h| + k as,

f(x) = a (x - h) + k, if (x - h) ≥ 0 and

= – a (x - h) + k, if (x - h) < 0

Absolute Value Function Graph

In this section, we will understand how to plot the graph of the common form of the absolute value function f(x) = |x| whose formula can also be expressed as f(x) = x, if x ≥ 0 and -x, if x < 0. Let us consider different points and determine the value of the function using the formula and plot them on a graph.

Absolute Value Equation

Now that we have understood the meaning of the absolute value function, now we will understand the meaning of the absolute value equation f(x) = a |x - h| + k and how the values of a, h, k affect the value of the function.

• The value of 'a' determines how the graph of f(x) stretches vertically
• The value of 'h' tells the horizontal shift
• The value of 'k' tells the vertical shift

The vertex of the absolute value equation f(x) = a |x - h| + k is given by (h, k). We can also find the vertex of f(x) = a |x - h| + k using the formula (x - h) = 0. On determining the value of x, we substitute the value into the equation to find the value of k.

Let us consider an example and find the vertex of an absolute value equation.

Example 1: Consider the modulus function f(x) = |x|. Find its vertex.

Solution: Compare the function f(x) = |x| with f(x) = a |x - h| + k. We have a = 1, h = k = 0. So, the vertex of the function is (h, k) = (0, 0).

Example 2: Find the vertex f(x) = |x - 7| + 2.

Solution: On comparing f(x) = |x - 7| + 2 with f(x) = a |x - h| + k, we have the vertex (h, k) = (7, 2).

We can find it using the formula. So, we have (x - 7) = 0

⇒ x = 7

Now, substitute x = 7 into the equation f(x) = |x - 7| + 2, we have

f(x) = |7 - 7| + 2

So, the vertex of absolute value equation f(x) = |x - 7| + 2 using the formula is (7, 2).

Graphing Absolute Value Functions

In this section, we will learn graphing absolute value functions of the form f(x) = a |x - h| + k. The graph of an absolute value function is always either 'V-shaped or inverted 'V-shaped depending upon the value of 'a' and the (h, k) gives the vertex of the graph. Let us plot the graph of two absolute value functions below.

f(x) = 2 |x + 2| + 1 and g(x) = -2 |x - 2| + 3

On comparing the two absolute value functions with the general form, a is positive in f(x), so it will open upwards and its vertex is (-2, 1). For g(x), the value of a = -2 which is negative, so the graph will open downwards and its vertex is (2, 3). The image below shows the graph of the absolute value functions f(x) and g(x).

Important Notes on Absolute Value Function

• The general form of the absolute value function is f(x) = a |x - h| + k, where (h, k) is the vertex of the graph.
• An absolute value function is a function in algebra where the variable is inside the absolute value bars.
• The graph of an absolute value function is always either 'V-shaped or inverted 'V-shaped depending upon the value of 'a'.

☛ Related Articles:

• Constant Function
• Inverse of a Function
• Graphing Functions

Absolute Value Function Examples

Example 1: Find the vertex of the absolute value function f(x) = -2 |x - 1| - 3

Solution: On comparing the function f(x) = -2 |x - 1| - 3 with f(x) = a |x - h| + k, the vertex of the function is (h, k) = (1, -3)

Answer: Vertex = (1, -3)

Example 2: Plot the graph of absolute value function f(x) = - |x + 2| - 3

Solution: As we can see, the value of a is -1 in f(x) = - |x + 2| - 3 which is negative. SO, the graph opens downwards and hence will be inverted V-shaped. The vertex of the graph is (h, k) = (-2, -3).

So, the graph of the given absolute value function f(x) = - |x + 2| - 3 is given by,

Example 3: Find the derivative of the absolute value function f(x) = |x|.

Solution: We can write f(x) = |x| as,

f(x) = x, if x ≥ 0 and

f(x) = -x, if x < 0

Now, we know that the derivative of x is 1 and the derivative of -x is -1. So, the derivative of f(x) = |x| is given by,

d(|x|)/dx = 1, if x > 0 and

= -1, if x < 0

Answer: d(|x|)/dx = 1, if x > 0 and -1, if x < 0.

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Absolute Value Function Questions

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FAQs on Absolute Value Function

An absolute value function is an important function in algebra that consists of the variable in the absolute value bars. The general form of the absolute value function is f(x) = a |x - h| + k, where (h, k) is the vertex of the function.

What is an Example of Absolute Value Function?

Some of the examples of absolute value functions are:

• g(x) = 2 |3x - 5| + 5
• f(x) = |-x - 9|
• f(x) = 3 |x|

How To Find the Vertex of an Absolute Value Function?

The general form of the absolute value function is f(x) = a |x - h| + k, where (h, k) is the vertex of the function. So, to find the vertex of the function, we compare the two equations and determine the values of h and k.

What Does the Value of k Do to the Absolute Value Function?

The value of 'k' in f(x) = a |x - h| + k tells us the vertical shift from the graph of f(x) = |x|. The graph moves upwards if k > 0 and moves downwards if k < 0.

Why is An Absolute Value Function Not Differentiable?

An absolute value function f(x) = a |x - h| + k is not differentiable at the vertex (h, k) because the left-hand limit and the right-hand limit of the function are not equal at the vertex.

Is an Absolute Value Function Even or Odd?

The absolute value function f(x) = |x| is an even function because f(x) = |x| = |-x| = f(-x) for all values of x.

How to Write an Absolute Value Function as a Piecewise Function?

We can write the absolute value function f(x) = |x| as a piecewise function as, f(x) = x, if x ≥ 0 and -x, if x < 0.

• EXPLORE Random Article

How to Solve Absolute Value Equations

Last Updated: January 31, 2023 References

This article was co-authored by wikiHow Staff . Our trained team of editors and researchers validate articles for accuracy and comprehensiveness. wikiHow's Content Management Team carefully monitors the work from our editorial staff to ensure that each article is backed by trusted research and meets our high quality standards. This article has been viewed 23,117 times.

Setting up the Problem

• For example, |9| = 9; |-9| = -(-9) = 9.

Calculating the Values

Check Your Work

Expert Q&A

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• Remember that absolute value bars are distinct from parentheses and function differently. Thanks Helpful 0 Not Helpful 0
• Once you've solved for any variables, remember to simplify absolute values accordingly. Thanks Helpful 0 Not Helpful 0

You Might Also Like

• ↑ http://tutorial.math.lamar.edu/Classes/Alg/SolveAbsValueEqns.aspx
• ↑ https://www.mathsisfun.com/numbers/absolute-value.html
• ↑ http://www.varsitytutors.com/high_school_math-help/solving-absolute-value-equations
• ↑ http://www.purplemath.com/modules/solveabs.htm
• ↑ https://www.khanacademy.org/math/algebra/absolute-value-equations-functions/absolute-value-equations/v/absolute-value-equations

About this article

To solve absolute value equations, first isolate the absolute value terms by moving anything outside of the vertical bars to the other side of the equation. Next, solve for the positive value of the equation by isolating the variable. Since the absolute variable can represent 2 numbers, then solve for the negative value by putting a negative sign outside the vertical bars. Then, move the negative by dividing both sides by -1 and solve for the variable. If you want to learn how to check your answers for an absolute value equation, keep reading the article! Did this summary help you? Yes No

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How to Write an Absolute-Value Equation That Has Given Solutions

How to Solve Absolute Value Equations With a Number on the Outside

You can denote absolute value by a pair of vertical lines bracketing the number in question. When you take the absolute value of a number, the result is always positive, even if the number itself is negative. For a random number ​ x ​, both the following equations are true: |-​ x ​| = ​ x ​ and |​ x ​| = ​ x ​. This means that any equation that has an absolute value in it has two possible solutions. If you already know the solution, you can tell immediately whether the number inside the absolute value brackets is positive or negative, and you can drop the absolute value brackets.

TL;DR (Too Long; Didn't Read)

Absolute value equations have two solutions. Plug in known values to determine which solution is correct, then rewrite the equation without absolute value brackets.

Consider the equality

To solve this, you have to set up two equalities and solve each separately.

Set up two separate (and unrelated) equations for ​ x ​ in terms of ​ y ​, being careful not to treat them as two equations in two variables:

This is solution for equation 1.

This is the solution for equation 2.

Because the original equation contained an absolute value, you're left with two relationships between ​ x ​ and ​ y ​ that are equally true. If you plot the above two equations on a graph, they will both be straight lines that intersect the origin. One has a slope of 4/3 while the other has a slope of 2/5.

If you have values for x and y for the above example, you can determine which of the two possible relationships between x and y is true, and this tells you whether the expression in the absolute value brackets is positive or negative.

Suppose you know the point ​ x ​ = 4, ​ y ​ = 10 is on the line. Plug these values into both equations.

Equation 2 is the correct one. You can now drop the absolute value brackets from the original equation and write instead:

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• SOS Math: Solving Equations Containing Absolute Values
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• You can also create an absolute value inequality simply by changing the equals sign to an inequality sign.

About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.

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