Skip to content Skip to navigation

Portal

You are here: Home » Content » Absolute Value Functions

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Absolute Value Functions

Module by: First Last. E-mail the author

Note: You are viewing an old style version of this document. The new style version is available here.

Figure 1: Distances in deep space can be measured in all directions. As such, it is useful to consider distance in terms of absolute values. (credit: "s58y"/Flickr)
The Milky Way.

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.

A General Note: Absolute Value Function:

The absolute value function can be defined as a piecewise function

f(x)=| x |={ x if x0 x if x<0 f(x)=| x |={ x if x0 x if x<0

Example 1

Problem 1

Determine a Number within a Prescribed Distance

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

Solution

We want the distance between x xand 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.

Figure 2
Number line describing the difference of the distance of 4 away from 5.

The distance from x xto 5 can be represented using the absolute value as | x5 |. | x5 |.We want the values of x xthat satisfy the condition | x5 |4. | x5 |4.

Analysis

Note that

4x5 x54 1x x9 4x5 x54 1x x9

So | x5 |4 | x5 |4is equivalent to 1x9. 1x9.

However, mathematicians generally prefer absolute value notation.

Try It:

Exercise 1

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

Solution

| x2 |3 | x2 |3

Example 2

Problem 1

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.

Solution

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 Rin ohms,

| R680 |34 | R680 |34

Try It:

Exercise 2

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

Solution

using the variable p pfor passing, | p80 |20 | p80 |20

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 3
Graph of an absolute function

Figure 4 shows the graph of y=2| x3 |+4. y=2| x3 |+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.

Figure 4
Graph of the different types of transformations for an absolute function.

Example 3

Problem 1

Writing an Equation for an Absolute Value Function

Write an equation for the function graphed in Figure 5.

Figure 5
Graph of an absolute function.
Solution

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.

Figure 6
Graph of two transformations for an absolute function at (3, -2).

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.

Figure 7
Graph of two transformations for an absolute function at (3, -2) and describes the ratios between the two different transformations.

From this information we can write the equation

f(x)=2|x3|2, treating the stretch as a vertical stretch, or f(x)=|2(x3)|2, treating the stretch as a horizontal compression. f(x)=2|x3|2, treating the stretch as a vertical stretch, or f(x)=|2(x3)|2, treating the stretch as a horizontal compression.
Analysis

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.

Q&A:

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 xand f(x). f(x).

f(x)=a|x3|2 f(x)=a|x3|2

Now substituting in the point (1, 2)

2=a| 13 |2 4=2a a=2 2=a| 13 |2 4=2a a=2

Try It:

Exercise 3

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

Solution

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

Q&A:

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).

Figure 8: (a) The absolute value function does not intersect the horizontal axis. (b) The absolute value function intersects the horizontal axis at one point. (c) The absolute value function intersects the horizontal axis at two points.
Graph of the different types of transformations for an absolute function.

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=| 2x6 |, 8=| 2x6 |,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.

2x6=8 or 2x6=8 2x=14 2x=2 x=7 x=1 2x6=8 or 2x6=8 2x=14 2x=2 x=7 x=1

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,

| x |=4, | 2x1 |=3 | 5x+2 |4=9 | x |=4, | 2x1 |=3 | 5x+2 |4=9

A General Note: Solutions to Absolute Value Equations:

For real numbers A Aand B, B,an equation of the form | A |=B, | A |=B,with B0, B0,will have solutions when A=B A=Bor A=B. A=B.If B<0, B<0,the equation | A |=B | A |=Bhas no solution.

How To:

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

  1. Isolate the absolute value term.
  2. Use | A |=B | A |=Bto write A=B A=Bor −A=B, −A=B, assuming B>0. B>0.
  3. Solve for x. x.

Example 4

Problem 1

Finding the Zeros of an Absolute Value Function

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

Solution
0=|4x+1|7 Substitute 0 for f(x). 7=|4x+1| Isolate the absolute value on one side of the equation. 7=4x+1 or 7=4x+1 Break into two separate equations and solve. 6=4x 8=4x x= 6 4 =1.5    x= 8 4 =2 0=|4x+1|7 Substitute 0 for f(x). 7=|4x+1| Isolate the absolute value on one side of the equation. 7=4x+1 or 7=4x+1 Break into two separate equations and solve. 6=4x 8=4x x= 6 4 =1.5    x= 8 4 =2

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

Figure 9
Graph an absolute function with x-intercepts at -2 and 1.5.

Try It:

Exercise 4

For the function f(x)=| 2x1 |3, f(x)=| 2x1 |3, find the values of x x such that f(x)=0. f(x)=0.

Solution

x=1 x=1or x=2 x=2

Q&A:

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+| 3x5 |=1. 2+| 3x5 |=1.

How To:

Given an absolute value equation, solve it.

  1. Isolate the absolute value term.
  2. Use | A |=B | A |=Bto write A=B A=Bor A=−B. A=−B.
  3. Solve for x. x.

Example 5

Problem 1

Solving an Absolute Value Equation

Solve 1=4| x2 |+2. 1=4| x2 |+2.

Solution

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

1=4| x2 |+2 1=4| x2 | 1 4 =| x2 | 1=4| x2 |+2 1=4| x2 | 1 4 =| x2 |

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.

Q&A:

In Example 5, if f(x)=1 f(x)=1and g(x)=4| x2 |+2 g(x)=4| x2 |+2were graphed on the same set of axes, would the graphs intersect?

No. The graphs of f fand g gwould not intersect, as shown in Figure 10. This confirms, graphically, that the equation 1=4| x2 |+2 1=4| x2 |+2 has no solution.

Figure 10
Graph of g(x)=4|x-2|+2 and f(x)=1.

Try It:

Exercise 5

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

Solution

f(0)=1, f(0)=1,so the graph intersects the vertical axis at (0,1). (0,1). f(x)=0 f(x)=0when x=5 x=5and x=1 x=1so the graph intersects the horizontal axis at (5,0) (5,0)and (1,0). (1,0).

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

|A|<B,|A|B,|A|>B,or |A|B, |A|<B,|A|B,|A|>B,or |A|B,

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 xthat 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

| x |<200or200<x<200  | x |<200or200<x<200 

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 xand 600 is less than 200. We represent the distance between x x and 600 as | x600 |. | x600 |.

|x600|<200    or    200<x600<200   200+600<x600+600<200+600                       400<x<800 |x600|<200    or    200<x600<200   200+600<x600+600<200+600                       400<x<800

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.

How To:

Given an absolute value inequality of the form | xA |B | xA |B for real numbers a aand b bwhere b bis positive, solve the absolute value inequality algebraically.

  1. Find boundary points by solving | xA |=B. | xA |=B.
  2. Test intervals created by the boundary points to determine where | xA |B. | xA |B.
  3. Write the interval or union of intervals satisfying the inequality in interval, inequality, or set-builder notation.

Example 6

Problem 1

Solving an Absolute Value Inequality

Solve |x5|4. |x5|4.

Solution

With both approaches, we will need to know first where the corresponding equality is true. In this case we first will find where | x5 |=4. | x5 |=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 | x5 |=4. | x5 |=4.

x5=4 x=9 or x5=4 x=1 x5=4 x=9 or x5=4 x=1

After determining that the absolute value is equal to 4 at x=1 x=1and 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:

x<1, 1<x<9, and  x>9. x<1, 1<x<9, and  x>9.

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.

Table 1
Interval test x x f(x) f(x) <4 <4or >4? >4?
x<1 x<1 0 | 05 |=5 | 05 |=5 Greater than
1<x<9 1<x<9 6 | 65 |=1 | 65 |=1 Less than
x>9 x>9 11 | 115 |=6 | 115 |=6 Greater than

Because 1x9 1x9is the only interval in which the output at the test value is less than 4, we can conclude that the solution to | x5 |4 | x5 |4is 1x9, 1x9,or [ 1,9 ]. [ 1,9 ].

To use a graph, we can sketch the function f(x)=| x5 |. f(x)=| x5 |.To help us see where the outputs are 4, the line g(x)=4 g(x)=4could also be sketched as in Figure 11.

Figure 11: Graph to find the points satisfying an absolute value inequality.
Graph of an absolute function and a vertical line, demonstrating how to see what outputs are less than the vertical line.

We can see the following:

  • The output values of the absolute value are equal to 4 at x=1 x=1and x=9. x=9.
  • The graph of f fis below the graph of g gon 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 1x9. 1x9.In interval notation, this would be the interval [ 1,9 ]. [ 1,9 ].
Analysis

For absolute value inequalities,

|xA|<C, |xA|>C, C<xA<C, xA<C or xA>C. |xA|<C, |xA|>C, C<xA<C, xA<C or xA>C.

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

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

|x5|4 4x54 Rewrite by removing the absolute value bars. 4+5x5+54+5 Isolate the x. 1x9 |x5|4 4x54 Rewrite by removing the absolute value bars. 4+5x5+54+5 Isolate the x. 1x9

Try It:

Exercise 6

Solve | x+2 |6. | x+2 |6.

Solution

4x8 4x8

How To:

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

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

Example 7

Problem 1

Using a Graphical Approach to Solve Absolute Value Inequalities

Given the function f(x)= 1 2 | 4x5 |+3, f(x)= 1 2 | 4x5 |+3, determine the x- x- values for which the function values are negative.

Solution

We are trying to determine where f(x)<0, f(x)<0, which is when 1 2  |4x5|+3<0. 1 2  |4x5|+3<0. We begin by isolating the absolute value.

1 2 |4x5|<3 Multiply both sides by –2, and reverse the inequality. |4x5|>6 1 2 |4x5|<3 Multiply both sides by –2, and reverse the inequality. |4x5|>6

Next we solve for the equality | 4x5 |=6. | 4x5 |=6.

4x5=6 4x5=6 4x5=6   or   4x=1 x= 11 4 x= 1 4 4x5=6 4x5=6 4x5=6   or   4x=1 x= 11 4 x= 1 4

Now, we can examine the graph of f fto 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.

Figure 12
Graph of an absolute function with x-intercepts at -0.25 and 2.75.

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.

x< 1 4 orx> 11 4 x< 1 4 orx> 11 4

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

Try It:

Exercise 7

Solve 2| k4 |6. 2| k4 |6.

Solution

k1 k1or k7; k7;in interval notation, this would be (,1][7,) (,1][7,)

Key Concepts

  • The absolute value function is commonly used to measure distances between points. See Example 1.
  • Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See Example 2.
  • The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See Example 3.
  • In an absolute value equation, an unknown variable is the input of an absolute value function.
  • If the absolute value of an expression is set equal to a positive number, expect two solutions for the unknown variable. See Example 4.
  • An absolute value equation may have one solution, two solutions, or no solutions. See Example 5.
  • An absolute value inequality is similar to an absolute value equation but takes the form | A |<B,| A |B,| A |>B,or | A |B. | A |<B,| A |B,| A |>B,or | A |B.It can be solved by determining the boundaries of the solution set and then testing which segments are in the set. See Example 6.
  • Absolute value inequalities can also be solved graphically. See Example 7.

Section Exercise

Verbal

Exercise 8

How do you solve an absolute value equation?

Solution

Isolate the absolute value term so that the equation is of the form |A|=B. |A|=B.Form one equation by setting the expression inside the absolute value symbol, A, A,equal to the expression on the other side of the equation, B. B.Form a second equation by setting A Aequal to the opposite of the expression on the other side of the equation, B. B.Solve each equation for the variable.

Exercise 9

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

Exercise 10

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?

Solution

The graph of the absolute value function does not cross the x x-axis, so the graph is either completely above or completely below the x x-axis.

Exercise 11

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

Exercise 12

How do you solve an absolute value inequality algebraically?

Solution

First determine the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to determine which values satisfy the inequality.

Algebraic

Exercise 13

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

Exercise 14

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

Solution

| x+4 |= 1 2 | x+4 |= 1 2

Exercise 15

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

Exercise 16

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.

Solution

|f(x)8|<0.03 |f(x)8|<0.03

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

Exercise 17

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

Exercise 18

|6x|=5 |6x|=5

Solution

{ 1,11 } { 1,11 }

Exercise 19

|5x2|=11 |5x2|=11

Exercise 20

|4x2|=11 |4x2|=11

Solution

{ 9 4 , 13 4 } { 9 4 , 13 4 }

Exercise 21

2|4x|=7 2|4x|=7

Exercise 22

3|5x|=5 3|5x|=5

Solution

{ 10 3 , 20 3 } { 10 3 , 20 3 }

Exercise 23

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

Exercise 24

5| x4 |7=2 5| x4 |7=2

Solution

{ 11 5 , 29 5 } { 11 5 , 29 5 }

Exercise 25

0=| x3 |+2 0=| x3 |+2

Exercise 26

2| x3 |+1=2 2| x3 |+1=2

Solution

{ 5 2 , 7 2 } { 5 2 , 7 2 }

Exercise 27

| 3x2 |=7 | 3x2 |=7

Exercise 28

| 3x2 |=7 | 3x2 |=7

Solution

No solution

Exercise 29

| 1 2 x5 |=11 | 1 2 x5 |=11

Exercise 30

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

Solution

{ 57,27 } { 57,27 }

Exercise 31

| 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.

Exercise 32

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

Solution

( 0,8 );( 6,0 ),( 4,0 ) ( 0,8 );( 6,0 ),( 4,0 )

Exercise 33

f(x)=4| x3 |+4 f(x)=4| x3 |+4

Exercise 34

f(x)=3| x2 |1 f(x)=3| x2 |1

Solution

( 0,7 ); ( 0,7 );no x x-intercepts

Exercise 35

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.

Exercise 36

| x2 |>10 | x2 |>10

Solution

(,8)(12,) (,8)(12,)

Exercise 37

2| v7 |442 2| v7 |442

Exercise 38

| 3x4 |8 | 3x4 |8

Solution

4 3 x4 4 3 x4

Exercise 39

| x4 |8 | x4 |8

Exercise 40

| 3x5 |13 | 3x5 |13

Solution

( , 8 3 ][ 6, ) ( , 8 3 ][ 6, )

Exercise 41

| 3x5 |13 | 3x5 |13

Exercise 42

| 3 4 x5 |7 | 3 4 x5 |7

Solution

( , 8 3 ][ 16, ) ( , 8 3 ][ 16, )

Exercise 43

| 3 4 x5 |+116 | 3 4 x5 |+116

Graphical

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

Exercise 44

y=|x1| y=|x1|

Solution
Graph of an absolute function with points at (-1, 2), (0, 1), (1, 0), (2, 1), and (3, 2).

Exercise 45

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

Exercise 46

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

Solution
Graph of an absolute function with points at (-2, 3), (-1, 2), (0, 1), (1, 2), and (2, 3).

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

Exercise 47

y=| x |2 y=| x |2

Exercise 48

y=| x | y=| x |

Solution
Graph of an absolute function.

Exercise 49

y=| x |2 y=| x |2

Exercise 50

y=| x3 |2 y=| x3 |2

Solution
Graph of an absolute function.

Exercise 51

f(x)=|x1|2 f(x)=|x1|2

Exercise 52

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

Solution
Graph of an absolute function.

Exercise 53

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

Exercise 54

f(x)=3| x2 |+3 f(x)=3| x2 |+3

Solution
Graph of an absolute function.

Exercise 55

f(x)=| 2x4 |3 f(x)=| 2x4 |3

Exercise 56

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

Solution
Graph of an absolute function.

Exercise 57

f(x)=| x1 |3 f(x)=| x1 |3

Exercise 58

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

Solution
Graph of an absolute function.

Exercise 59

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

Technology

Exercise 60

Use a graphing utility to graph f(x)=10|x2| f(x)=10|x2| on the viewing window [ 0,4 ]. [ 0,4 ]. Identify the corresponding range. Show the graph.

Solution

range: [ 0,20 ] [ 0,20 ]

Graph of an absolute function.

Exercise 61

Use a graphing utility to graph f(x)=100|x|+100 f(x)=100|x|+100on 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.

Exercise 62

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

Solution

x- x- intercepts:

Graph of an absolute function.

Exercise 63

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

Extensions

For the following exercises, solve the inequality.

Exercise 64

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

Solution

(,) (,)

Exercise 65

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.

Exercise 66

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

Solution

There is no solution for a athat will keep the function from having a y y-intercept. The absolute value function always crosses the y y-intercept when x=0. x=0.

Real-World Applications

Exercise 67

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 xrepresents the distance from city B to city A, express this using absolute value notation.

Exercise 68

The true proportion p pof 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.

Solution

| p0.08 |0.015 | p0.08 |0.015

Exercise 69

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 xfor the score.

Exercise 70

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

Solution

| x5.0 |0.01 | x5.0 |0.01

Exercise 71

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 xinches, express the tolerance using absolute value notation.

Glossary

absolute value equation:
an equation of the form | A |=B, | A |=B,with B0; B0;it will have solutions when A=B A=Bor A=B A=B
absolute value inequality:
a relationship in the form | A |<B,| A |B,| A |>B,or | A |B | A |<B,| A |B,| A |>B,or | A |B

Content actions

Download module as:

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks