#### Solved Examples and Worksheet for Solving and Graphing Absolute Value Inequalities

Q1A rope is cut into 7 equal pieces. A small piece of 2 ft is again cut from each of the 7 pieces. Choose the correct variable expression for the new length of each piece.

A. (x7 - 2) ft
B. (x2 - 7) ft
C. (2 - x7) ft
D. (x7 + 2) ft

Step: 1
Let x be the length of the original rope.
Step: 2
Length of each of the 7 pieces = x7 ft
Step: 3
The variable expression for the new length of each piece is (x7- 2) ft.
[A length of 2 ft is cut from each of the 7 pieces.]
Correct Answer is :   (x7 - 2) ft
Q2Identify a variable expression for the word phrase '13 more than 8 times a number'.
A. 13n + 8
B. 8n + 13
C. n + 91
D. 8n - 13

Step: 1
Let n be a number.
Step: 2
13 more than 8 times a number = 8n + 13.
[Write the expression.]
Step: 3
So, the variable expression for the phrase is 8n + 13.
Correct Answer is :   8n + 13
Q3Write a variable expression for the phrase, 'the number of days in n weeks'.
A. n - 7
B. 7n
C. 7 + n
D. 7n

Step: 1
1 week = 7 days.
Step: 2
Number of days in n weeks = n × number of days in a week = n × 7 = 7n.
Step: 3
The variable expression for this situation is 7n.
Q4The amount of orange juice present in a pitcher is 1 liter less than twice the amount of juice present in the other pitcher. Choose an algebraic expression to represent this situation.
A. 2 + n - 1
B. n2 - 1
C. 2n + 1
D. 2n - 1

Step: 1
Twice of an amount of juice will be equal to 2 × n or 2n.
Step: 2
One liter less than this amount will be equal to 2n - 1.
Step: 3
2n - 1 is the correct answer.
Correct Answer is :   2n - 1
Q5The amount of sugar present in a jar is 10g less than twice the amount of sugar present in the other jar. Choose an algebraic expression to represent this situation.
A. 2n - 10
B. 2 + n - 10
C. 2n + 10
D. n2 - 10

Step: 1
Twice of an amount of sugar will be equal to 2 × n or 2n.
Step: 2
Ten grams less than this amount will be equal to 2n - 10.
Step: 3
Correct Answer is :   2n - 10
Q6Lindsay's salary was $y 3 years ago. Now, she gets 4 times her previous salary and she spends$1512. Calculate her savings.

A. $(1512 + 4y) B.$(1512 - y)
C. $(1512 - 4y) D.$(4y - 1512)

Step: 1
Lindsay's present salary is $4y. Step: 2 Lindsay's savings =$(4y - 1512)
[Savings = Income - Expenditure.]
Correct Answer is :   \$(4y - 1512)
Q7Paul and Matt fire 46 shots each to hit a target. Each person gets p points for a hit and loses q points for a miss. Paul hits the mark 40 times and Matt hits the mark 38 times. Identify the variable expressions for their individual scores.

A. 40p - 6, 38p - 8
B. 40p, 38q
C. 40p - 6q, 38p - 8q
D. 38p, 40q

Step: 1
Points scored by each person = Number of hits × p - Number of missed hits × q
Step: 2
Number of hits that Paul missed = (46 - 40) = 6
[Subtract.]
Step: 3
The variable expression for the points scored by Paul is 40p - 6q.
Step: 4
Number of hits that Matt missed = (46 - 38) = 8
[Subtract.]
Step: 5
The variable expression for the points scored by Matt is 38p - 8q.
Step: 6
So, the variable expressions for the points scored by Paul and Matt are 40p - 6q and 38p - 8q.
Correct Answer is :   40p - 6q, 38p - 8q
Q8Which of the following is the correct choice for the statement, "y is divided by a number 38"?

A. y - 38
B. y + 38
C. y38
D. 38y

Step: 1
y is divided by a number 38.
[In words.]
Step: 2
y38
[Variable expression.]
Step: 3
Variable expression for the word phrase is y38.
Q9Which of the following choices represents the phrase "3 added to 5 times x is equal to 19"?

A. 5x - 3 = 19
B. 5(x + 3) = 19
C. x3 + 3 = 19
D. 3 + 5x = 19

Step: 1
5 times x = 5x
Step: 2
3 added to 5x = 3 + 5x
Step: 3
3 added to 5 times x is equal to 19 is 3 + 5x = 19.
Correct Answer is :   3 + 5x = 19
Q10Choose an algebraic sentence for " Seven times the quantity of a number decreased by ten is equal to 186 more than the number". Let c be the number.

A. 7(c - 10) = 186 + c
B. 7(c - 10) > 186
C. 7c - 10 = 186c
D. 7c + 10 = 186 + c

Step: 1
Seven times the quantity of a number decreased by ten = 7(c - 10).
Step: 2
186 more than the number = 186 + c.
Step: 3
The algebraic sentence for the given word sentence: 7(c - 10) = 186 + c.
Correct Answer is :   7(c - 10) = 186 + c
Q11Which of the following choices represents the word phrase "4 added to 4 times x is equal to 19"?

A. x4 + 4 = 19
B. 4(x + 4) = 19
C. 4 + 4x = 19
D. 4x - 4 = 19

Step: 1
4 times x = 4x
Step: 2
4 added to 4x = 4 + 4x
Step: 3
4 added to 4 times x is equal to 19 is 4 + 4x = 19
Correct Answer is :   4 + 4x = 19
Q12Choose the graph for the inequality, | w + 52 | - 4 > 5.

A. Graph A
B. Graph B
C. Graph C
D. Graph D

Step: 1
| w + 52 | - 4 > 5
Step: 2
| w + 52 | > 9
Step: 3
w + 5 2 > 9 or w + 52 < -9
[Write the equivalent disjunction.]
Step: 4
(w + 5) > 18 or (w + 5) < -18
[Multiply throughout by 2.]
Step: 5
w > 13 or w < - 23
Step: 6
The solution set is {w: w < - 23 or w > 13}
Step: 7
The graphical representation is as follows:
Correct Answer is :   Graph C
Q13Choose the graph for the inequality, 5 < | x - 4 |.

A. Graph A
B. Graph B
C. Graph C
D. Graph D

Step: 1
5 < | x - 4 |
Step: 2
5 < (x - 4) or - 5 > (x - 4)
[Write the equivalent disjunction.]
Step: 3
9 < x or - 1 > x
Step: 4
That is x < - 1 or x > 9
Step: 5
The solution set is {x: x < - 1 or x > 9}
Step: 6
The graphical representation is as follows:
Correct Answer is :   Graph C
Q14Choose the graph for the inequality, | 5 + 2(x - 1) | ≤ 1.

A. Graph B
B. Graph A
C. Graph D
D. Graph C

Step: 1
| 5 + 2(x - 1) | ≤1
Step: 2
5 + 2(x - 1) ≤ 1 and 5 + 2(x -1) ≥ - 1
Step: 3
2(x - 1) ≤- 4 and 2(x - 1) ≥ - 6
[Subtracting 5 from the two sides of the equation.]
Step: 4
(x - 1) ≤- 2 and (x - 1) ≥ - 3
[Divide throughout by 2.]
Step: 5
x ≤ - 1 and x ≥ - 2
[Add 1 to both sides of the equation.]
Step: 6
The solution set is {x: - 2 ≤ x ≤- 1}
Step: 7

Correct Answer is :   Graph B
Q15Choose an inequality for the graph shown.

A. | x + 1 | ≤ 2
B. | x + 1 | ≥ 2
C. | x + 1 | > 2
D. | x + 1 | < 2

Step: 1
The graph shows the values between - 3 and 1 including the points -3, 1.
Step: 2
-3 ≤ x ≤ 1
Step: 3
-2 ≤ x + 1 ≤ 2