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

Key concepts

  • Applied problems, such as ranges of possible values, can also be solved using the absolute value function. See [link] .
  • The graph of the absolute value function resembles a letter V. It has a corner point at which the graph changes direction. See [link] .
  • 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 [link] .

Section exercises

Verbal

How do you solve an absolute value equation?

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

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

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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?

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

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

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Algebraic

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

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

| x + 4 | = 1 2

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

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

| f ( x ) 8 | < 0.03

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

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

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f ( x ) = 3 | x 2 | 1

( 0 , 7 ) ; no x -intercepts

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

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f ( x ) = 5 | x + 2 | + 15

( 0 , 5 ) , ( 1 , 0 ) , ( 5 , 0 )

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f ( x ) = 2 | x 1 | 6

( 0 , 4 ) , ( 4 , 0 ) , ( 2 , 0 )

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f ( x ) = | 2 x + 1 | 13

( 0 , 12 ) , ( 6 , 0 ) , ( 7 , 0 )

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f ( x ) = | x 9 | + 16

( 0 , 7 ) , ( 25 , 0 ) , ( 7 , 0 )

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Graphical

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

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

y = | x 3 | 2

Graph of an absolute function.
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f ( x ) = | x 1 | 2

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

Graph of an absolute function.
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f ( x ) = 2 | x + 3 | + 1

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

Graph of an absolute function.
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f ( x ) = | 2 x 4 | 3

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

Graph of an absolute function.
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f ( x ) = | x 1 | 3

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f ( x ) = | x + 4 | 3

Graph of an absolute function.
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f ( x ) = 1 2 | x + 4 | 3

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Technology

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

range: [ 0 , 20 ]

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

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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

x - intercepts:

Graph of an absolute function.
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f ( x ) = 4 × 10 9 | x ( 5 × 10 9 ) | + 2 × 10 9

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Extensions

For the following exercises, solve the inequality.

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

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

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

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

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The true proportion 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.

| p 0.08 | 0.015

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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 for the score.

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

| x 5.0 | 0.01

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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 inches, express the tolerance using absolute value notation.

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Questions & Answers

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the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
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An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
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lim x to infinity e^1-e^-1/log(1+x)
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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