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# Absolute Value Function Worksheet - Page 3

Absolute Value Function Worksheet
• Page 3
21.
Determine whether the given statement is true or false: If $a$ > 0, then |7 - $a$| = 7 - $a$
 a. False b. True

#### Solution:

If a > 0, then |7 - a| = 7 - a

Take one example, a = 16

|7 - 16| = 7 - 16
[Substitute the values.]

|- 9| = - 9
[Simplify.]

9 = - 9, which is not true.

Correct answer : (1)
22.
Determine whether the given statement is True or False: If $a$ > 0, then |- $a$ - 2| = 2 + $a$
 a. False b. True

#### Solution:

If a > 0, then |- a - 2| = 2 + a

|- (a + 2)| = 2 + a

Since a > 0, a + 2 is also greater than zero.

a + 2 = a + 2, which is true.
[|- (a + 2)| = a + 2.]

Correct answer : (2)
23.
Determine whether the given sentence is True or False: If $c$ < 0, then |- $c$ - 8| = $c$ + 8.
 a. True b. False

#### Solution:

If c < 0, then |- c - 8| = c + 8

Take one example c = - 17

|- (- 17) - 8| = - 17 + 8
[Substitute the values.]

|17 - 8| = - 17 + 8

9 = - 9, which is not true.

Correct answer : (2)
24.
Solve for ' $b$ ', |7$b$ + 10| = - 11.
 a. $b$ = 11 b. $b$ = - 7 c. No solution d. $b$ = $\frac{1}{7}$

#### Solution:

The absolute value will never be negative.

Hence, there is no solution for |7b + 10| = - 11.

Correct answer : (3)
25.
Solve for ' $a$ ', |5$a$ - 8| = 10$a$ - 4
 a. $a$ = $\frac{4}{5}$ or $a$ = - $\frac{4}{5}$ b. $a$ = - $\frac{4}{5}$ only c. $a$ = $\frac{4}{5}$ only d. None of the above

#### Solution:

|5a - 8| = 10a - 4

5a - 8 = 10a - 4 or 5a - 8 = - (10a - 4)

- 8 = 5a - 4 or - 8 = - 15a + 4
[Subtracting 5a from the two sides of the equation.]

- 4 = 5a or - 12 = - 15x
[Group the like terms.]

- 45 = a or 45 = a
[Simplify.]

a = - 45 or a = 45
[Reflexive property.]

Check the answer: a = - 45

|5 (- 45) - 8| = 10 (- 45) - 4

|- 12 | = - 8 - 4

12 = - 12, which is not True.

So, a = - 45 is not the solution.

Check the answer a = 45

|5 (45) - 8| = 10 (45) - 4

|4 - 8| = 8 - 4

4 = 4, which is true

So, a = 45 is the solution.

Correct answer : (3)
26.
Solve for $z$: - | 5 - 6$z$| = 10
 a. $z$ = $\frac{35}{2}$ b. Real numbers c. No solution d. $z$ = - $\frac{5}{2}$

#### Solution:

The absolute value will never be negative.

|5 - 6z| is always positive.

So, - |5 - 6z| will always be negative.

So, the given equation has no solution.

Correct answer : (3)
27.
Solve: 6 | $y$ + 11| = 18$y$ + 48
 a. $y$ = $\frac{3}{2}$ only b. $y$ = - $\frac{19}{4}$ or $y$ = $\frac{3}{2}$ c. $y$ = - $\frac{19}{2}$ only d. No solution

#### Solution:

6 |y + 11| = 18y + 48

|y + 11| = 3y + 8
[Divide throughout by 6 .]

y + 11 = 3y + 8 or y + 11 = - (3y + 8)
[Write it as disjunction.]

- 2y = - 3 or 4y = - 19
[Group the variables and constants seperately.]

y = 32 or y = - 194
[Evaluate.]

Check the answer.

First check for y = 32

6 |32 + 11| = 18 (32) + 48

6 |252| = 27 + 48

75 = 75, which is true.

Check for y = - 194

6 | - 194 + 11| = 18 (- 194) + 48

6 | 254| = - 85.50 + 48

37.50 = - 37.50, which is not true.

So, y = - 194 is not a solution.

The solution is y = 32.

Correct answer : (1)
28.
Solve: 9 |9 - 8$y$| = 36$y$ - 108.
 a. No solution b. $y$ = - $\frac{3}{4}$ only c. $y$ = $\frac{21}{12}$ or y = - $\frac{3}{4}$ d. $y$ = $\frac{21}{12}$ only

#### Solution:

9|9 - 8y| = 36y - 108

|9 - 8y| = 4y - 12
[divide by 9 .]

9 - 8y = 4y - 12 or 9 - 8y = - (4y - 12)
[Write it as a disjunction.]

- 12y = - 21 or - 4y = 3
[Simplify.]

y = 2112 or y = - 34

Check for y = 2112

9 |9 - 8 (2112)| = 36(2112) - 108
[Substitute the values.]

9 |(108 - 168)12| = 75612 - 108

9 |(108 - 168)12| = (756 - 1296)12

45 = -45 , which is not true.

Hence, y = 138 is not the solution.

Check for y = - 34

9 |9 - 8 (- 34)| = 36 (- 34) - 108

9 |9 + (244)| = - 1084 - 108

135 = - 135, which is not true.

Hence, y = - 34 is also not the solution.

Correct answer : (1)
29.
Solve and check $\frac{1}{3}$| 2$c$ + 3| = 5$c$ + 2.
 a. $c$ = - $\frac{3}{13}$ only b. No solution c. $c$ = - $\frac{9}{17}$or $x$ = - $\frac{3}{13}$ d. $c$ = - $\frac{3}{17}$only

#### Solution:

13|2c + 3| = 5c + 2

|2c + 3| = 15c + 6
[Multiply throughout by 3.]

2c + 3 = 15c + 6 or 2c + 3 = - (15c + 6)
[Write it as a disjunction.]

- 13c = 3 or 17c = - 9

c = - 313 or c = - 917

Check the answer for c = - 313

13 |2 (- 313) + 3| = 5 (- 313) + 2

13|(- 6 + 39)13 | = (- 15 + 26)13

11 / 13 = 11 / 13, which is true.

So, c = - 313 is the solution.

Check the answer: c = - 917

1 / 3 |2(- 9 / 17) + 3| = 5(- 9 / 17) + 2

1 / 3|(- 18 + 51)17 | = (- 45 + 34)17

11 / 17 = -11 / 17, which is not true.

So, c = - 917 is not the solution.

Hence, c = - 313 is only the solution.

Correct answer : (1)
30.
Solve for $x$ and check: |c$x$| - $a$ = $b$; $a$, $b$ and $c$ are positive real numbers.
 a. $x$ = only b. $x$ = only c. No solution d. $x$ = or $x$ =

#### Solution:

|cx| - a = b

|cx| = a + b
[Add 'a' to both sides of the equation.]

cx = a + b or cx = - (a + b)

x = (a + b)c or x = - (a + b)c

Check the answer: x = (a + b)c

|c (a + bc)| - a = b

|a + b| - a = b

a + b - a = b
[a + b is a positive real number.]

b = b, which is true.

So, x = (a + b)c is the solution.

Check the answer: x = (- a - b)c

|c (- a - bc)| - a = b

|- (a + b)| - a = b
[Use |- x| = | x| = x, x is a positive real number.]

a + b - a = b

b = b, which is true.

So, x = - a - bc is the solution.

Correct answer : (4)

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