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1.6E: Exercises

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Section Exercise

Verbal

Exercise 1.6.1

How do you solve an absolute value equation?

Answer:

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.

Exercise 1.6.2

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

Exercise 1.6.3

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?

Answer:

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.

Exercise 1.6.4

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

Exercise 1.6.5

How do you solve an absolute value inequality algebraically?

Answer:

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 1.6.6

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

Exercise 1.6.7

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

Answer:

$$|x+4|= \frac{1}{2}$$

Exercise 1.6.8

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

Exercise 1.6.9

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 using absolute value notation.

Answer:

$$|f(x)−8|<0.03$$

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

Exercise 1.6.10

$$|x+3|=9$$

Exercise 1.6.11

$$|6−x|=5$$

Answer:

$${1,11}$$

Exercise 1.6.12

$$|5x−2|=11$$

Exercise 1.6.13

$$|4x−2|=11$$

Answer:

$$\{\frac{9}{4}, \frac{13}{4}\}$$

Exercise 1.6.14

$$2|4−x|=7$$

Exercise 1.6.15

$$3|5−x|=5$$

Answer:

$$\{\frac{10}{3},\frac{20}{3}\}$$

Exercise 1.6.16

$$3|x+1|−4=5$$

Exercise 1.6.17

$$5|x−4|−7=2$$

Answer:

$$\{\frac{11}{5}, \frac{29}{5}\}$$

Exercise 1.6.18

$$0=−|x−3|+2$$

Exercise 1.6.19

$$2|x−3|+1=2$$

Answer:

$$\{\frac{5}{2}, \frac{7}{2}\}$$

Exercise 1.6.20

$$|3x−2|=7$$

Exercise 1.6.21

$$|3x−2|=−7$$

Answer:

No solution

Exercise 1.6.22

$$|\frac{1}{2}x−5|=11$$

Exercise 1.6.23

$$| \frac{1}{3}x+5|=14$$

Answer:

$$\{−57,27\}$$

Exercise 1.6.24

$$−|\frac{1}{3}x+5|+14=0$$

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

Exercise 1.6.25

$$f(x)=2|x+1|−10$$

Answer:

$$(0,−8)$$; $$(−6,0)$$, $$(4,0)$$

Exercise 1.6.26

$$f(x)=4|x−3|+4$$

Exercise 1.6.27

$$f(x)=−3|x−2|−1$$

Answer:

$$(0,−7)$$; no x-intercepts

Exercise 1.6.28

$$f(x)=−2|x+1|+6$$

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

Exercise 1.6.29

$$| x−2 |>10$$

Answer:

$$(−\infty,−8)\cup(12,\infty)$$

Exercise 1.6.30

$$2|v−7|−4\geq42$$

Exercise 1.6.31

$$|3x−4|\geq8$$

Answer:

$$−\dfrac{4}{3}{\leq}x\leq4$$

Exercise 1.6.32

$$|x−4|\geq8$$

Exercise 1.6.33

$$|3x−5|\geq-13$$

Answer:

$$(−\infty,− \frac{8}{3}]\cup\left[6,\infty\right)$$

Exercise 1.6.34

$$|3x−5|\geq−13$$

Exercise 1.6.35

$$|\frac{3}{4}x−5|\geq7$$

Answer:

$$(-\infty,-\frac{8}{3}]\cup\left[16,\infty\right)$$

Exercise 1.6.36

$$|\frac{3}{4}x−5|+1\leq16$$

Graphical

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

Exercise 1.6.37

$$y=|x−1|$$

Answer:

Exercise 1.6.38

$$y=|x+1|$$

Exercise 1.6.39

$$y=|x|+1$$

Answer:

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

Exercise 1.6.40

$$y=|x|−2$$

Exercise 1.6.41

$$y=−|x|$$

Answer:

Exercise 1.6.42

$$y=−|x|−2$$

Exercise 1.6.43

$$y=−|x−3|−2$$

Answer:

Exercise 1.6.44

$$f(x)=−|x−1|−2$$

Exercise 1.6.45

$$f(x)=−|x+3|+4$$

Answer:

Exercise 1.6.46

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

Exercise 1.6.47

$$f(x)=3|x−2|+3$$

Answer:

Exercise 1.6.48

$$f(x)=|2x−4|−3$$

Exercise 1.6.49

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

Answer:

Exercise 1.6.50

$$f(x)=−|x−1|−3$$

Exercise 1.6.51

$$f(x)=−|x+4|−3$$

Answer:

Exercise 1.6.52

$$f(x)=\frac{1}{2}|x+4|−3$$

Technology

Exercise 1.6.53

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

Answer:

range: $$[0,20]$$

Exercise 1.6.54

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

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

Exercise 1.6.55

$$f(x)=−0.1|0.1(0.2−x)|+0.3$$

Answer:

x-intercepts:

Exercise 1.6.56

$$f(x)=4 \times10^{9}|x−(5 \times 10^9)|+2 \times10^9$$

Extensions

For the following exercises, solve the inequality.

Exercise 1.6.57

$$|−2x− \frac{2}{3}(x+1)|+3>−1$$

Answer:

$$(−\infty,\infty)$$

Exercise 1.6.58

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

Exercise 1.6.59

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

Answer:

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 $$x=0$$.

Real-World Applications

Exercise 1.6.60

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.

Exercise 1.6.61

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.

Answer:

$$|p−0.08|\leq0.015$$

Exercise 1.6.62

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.

Exercise 1.6.63

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.

Answer:

$$|x−5.0|\leq0.01$$

Exercise 1.6.64

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.

1.6E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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