1
What is the length of the solution interval for |x + 34| < 7?
Medium
Explanation|x + 34| < 7 describes the interval (-41, -27), which has length -27 − (-41) = 14.
Explanation|40| − |-17| = 40 − 17 = 23. (Note this is generally different from |40 − (-17)|.)
3
How many solutions does |x + 14| = −16 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x + 14| = −16 has no solution.
4
What is the solution set of |x + 24| < 26?
Medium
Explanation|x + 24| < 26 means x is within 26 of -24, giving -50 < x < 2.
5
What are the solutions to |x + 19| = 13?
Medium
Explanation|x + 19| = 13 means x is 13 away from -19, so x = -19 + 13 = -6 or x = -19 − 13 = -32.
6
How many solutions does |x − 20| = −15 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x − 20| = −15 has no solution.
7
What is the solution set of |x − 16| < 12?
Medium
Explanation|x − 16| < 12 means x is within 12 of 16, giving 4 < x < 28.
8
What is the midpoint between -35 and -54 on the number line?
Medium
ExplanationThe midpoint of -35 and -54 is (-35 + -54) / 2 = -89/2 = -89/2.
9
What is the length of the solution interval for |x − 40| < 25?
Medium
Explanation|x − 40| < 25 describes the interval (15, 65), which has length 65 − (15) = 50.
10
What is the midpoint between -43 and 2 on the number line?
Medium
ExplanationThe midpoint of -43 and 2 is (-43 + 2) / 2 = -41/2 = -41/2.
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11
Which of the following numbers lies between 24 and 58?
Medium
Explanation57 lies strictly between 24 and 58, while the other options fall outside that range.
Explanation|-126 ÷ -14| = |-126| ÷ |-14| = 126 ÷ 14 = 9.
13
What is the solution set of |x + 3| < 15?
Medium
Explanation|x + 3| < 15 means x is within 15 of -3, giving -18 < x < 12.
14
Among -74, 16, -18, and -61, which is closest to 0 on the number line?
Medium
ExplanationThe number closest to 0 has the smallest absolute value. |16| = 16, which is the smallest among the options.
15
What are the solutions to |x| + 5 = 29?
Medium
Explanation|x| + 5 = 29 means |x| = 24, so x = 24 or x = -24.
16
What is the solution set of |x + 12| < 21?
Medium
Explanation|x + 12| < 21 means x is within 21 of -12, giving -33 < x < 9.
17
What is the solution set of |x + 4| > 21?
Medium
Explanation|x + 4| > 21 means x is farther than 21 from -4, giving x < -25 or x > 17.
18
What is the solution set of |x − 18| < 11?
Medium
Explanation|x − 18| < 11 means x is within 11 of 18, giving 7 < x < 29.
19
What are the solutions to |x + 38| = 5?
Medium
Explanation|x + 38| = 5 means x is 5 away from -38, so x = -38 + 5 = -33 or x = -38 − 5 = -43.
20
What are the solutions to |x − 31| = 2?
Medium
Explanation|x − 31| = 2 means x is 2 away from 31, so x = 31 + 2 = 33 or x = 31 − 2 = 29.
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21
How many solutions does |x − 29| = −4 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x − 29| = −4 has no solution.
22
What are the solutions to |x| + 1 = 10?
Medium
Explanation|x| + 1 = 10 means |x| = 9, so x = 9 or x = -9.
23
What are the solutions to |x + 11| = 13?
Medium
Explanation|x + 11| = 13 means x is 13 away from -11, so x = -11 + 13 = 2 or x = -11 − 13 = -24.
24
What are the solutions to |9x − 81| = 18?
Medium
Explanation9x − 81 = 18 or 9x − 81 = −18, which solve to x = 11 and x = 7.
25
What is the length of the solution interval for |x − 23| < 19?
Medium
Explanation|x − 23| < 19 describes the interval (4, 42), which has length 42 − (4) = 38.
26
What is the solution set of |x − 24| > 6?
Medium
Explanation|x − 24| > 6 means x is farther than 6 from 24, giving x < 18 or x > 30.
27
What is the midpoint between -6 and -23 on the number line?
Medium
ExplanationThe midpoint of -6 and -23 is (-6 + -23) / 2 = -29/2 = -29/2.
ExplanationAbsolute value gives the distance from 0, which is always non-negative, so |-18| = 18.
29
Which is greater: |70| or |42|?
Medium
Explanation|70| = 70 and |42| = 42, so |70| is greater.
30
How many solutions does |x − 24| = −12 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x − 24| = −12 has no solution.
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31
For a = -25 and b = 18, is |a + b| equal to |a| + |b|, or strictly less than |a| + |b|?
Medium
Explanation|a + b| = 7 and |a| + |b| = 43. Since a and b have opposite signs, the two are not equal, with |a+b| smaller.
32
What is the solution set of |x − 0| > 2?
Medium
Explanation|x − 0| > 2 means x is farther than 2 from 0, giving x < -2 or x > 2.
33
Which is greater: |-32| or |-66|?
Medium
Explanation|-32| = 32 and |-66| = 66, so |-66| is greater.
34
Among 70, 46, -14, and -23, which is closest to 0 on the number line?
Medium
ExplanationThe number closest to 0 has the smallest absolute value. |-14| = 14, which is the smallest among the options.
35
What is the distance between -95 and 79 on the number line?
Medium
ExplanationThe distance between two points on a number line is |-95 − 79| = 174.
36
How many solutions does the equation |x − 8| = 22 have?
Medium
Explanation|x − 8| = 22: since 22 is positive, there are two solutions.
37
How many solutions does |x − 15| = −20 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x − 15| = −20 has no solution.
38
What is the distance between -33 and -99 on the number line?
Medium
ExplanationThe distance between two points on a number line is |-33 − (-99)| = 66.
39
Which of the following numbers lies between 28 and 67?
Medium
Explanation51 lies strictly between 28 and 67, while the other options fall outside that range.
40
What are the solutions to |x + 21| = 1?
Medium
Explanation|x + 21| = 1 means x is 1 away from -21, so x = -21 + 1 = -20 or x = -21 − 1 = -22.
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41
What are the solutions to |x + 11| = 1?
Medium
Explanation|x + 11| = 1 means x is 1 away from -11, so x = -11 + 1 = -10 or x = -11 − 1 = -12.
42
Among 31, -22, 6, and 30, which is closest to 0 on the number line?
Medium
ExplanationThe number closest to 0 has the smallest absolute value. |6| = 6, which is the smallest among the options.
43
What are the solutions to |x| + 24 = 26?
Medium
Explanation|x| + 24 = 26 means |x| = 2, so x = 2 or x = -2.
44
Among -67, 84, 75, and -52, which is closest to 0 on the number line?
Medium
ExplanationThe number closest to 0 has the smallest absolute value. |-52| = 52, which is the smallest among the options.
45
If |x| = |-27|, what are the possible values of x?
Medium
Explanation|x| = |-27| = 27 means x is 27 units from 0, so x = 27 or x = -27.
46
How many integers satisfy |x + 21| < 18?
Medium
Explanation|x + 21| < 18 means -39 < x < -3. The integers strictly between -39 and -3 number 35.
Explanation|13 × -19| = |13| × |-19| = 13 × 19 = 247.
Explanation|9 × 18| = |9| × |18| = 9 × 18 = 162.
49
Among 66, 62, 17, and 26, which is closest to 0 on the number line?
Medium
ExplanationThe number closest to 0 has the smallest absolute value. |17| = 17, which is the smallest among the options.
50
What are the solutions to |2x − 8| = 18?
Medium
Explanation2x − 8 = 18 or 2x − 8 = −18, which solve to x = 13 and x = -5.
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51
Among -27, -13, 89, and -39, which is closest to 0 on the number line?
Medium
ExplanationThe number closest to 0 has the smallest absolute value. |-13| = 13, which is the smallest among the options.
52
What is the distance between -85 and 21 on the number line?
Medium
ExplanationThe distance between two points on a number line is |-85 − 21| = 106.
Explanation|-12| − |-15| = 12 − 15 = -3. (Note this is generally different from |-12 − (-15)|.)
Explanation|-12 × -15| = |-12| × |-15| = 12 × 15 = 180.
55
How many solutions does |x − 40| = −21 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x − 40| = −21 has no solution.
56
How many solutions does |x − 9| = −26 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x − 9| = −26 has no solution.
57
What is the solution set of |x − 22| > 20?
Medium
Explanation|x − 22| > 20 means x is farther than 20 from 22, giving x < 2 or x > 42.
58
What is the midpoint between 7 and 30 on the number line?
Medium
ExplanationThe midpoint of 7 and 30 is (7 + 30) / 2 = 37/2 = 37/2.
59
Which is greater: |-72| or |23|?
Medium
Explanation|-72| = 72 and |23| = 23, so |-72| is greater.
60
Among 55, -65, -32, and 53, which is closest to 0 on the number line?
Medium
ExplanationThe number closest to 0 has the smallest absolute value. |-32| = 32, which is the smallest among the options.
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Explanation|16 × -9| = |16| × |-9| = 16 × 9 = 144.
62
Which of the following numbers lies between -4 and 25?
Medium
Explanation23 lies strictly between -4 and 25, while the other options fall outside that range.
63
What are the solutions to |x + 20| = 1?
Medium
Explanation|x + 20| = 1 means x is 1 away from -20, so x = -20 + 1 = -19 or x = -20 − 1 = -21.
64
Among -76, -6, 59, and -75, which is closest to 0 on the number line?
Medium
ExplanationThe number closest to 0 has the smallest absolute value. |-6| = 6, which is the smallest among the options.
65
Which is farther from 21 on the number line: -60 or -3?
Medium
ExplanationDistance from 21: |-60 − 21| = 81, and |-3 − 21| = 24. So -60 is farther.
66
What is the distance between 18 and 65 on the number line?
Medium
ExplanationThe distance between two points on a number line is |18 − 65| = 47.
67
What are the solutions to |6x + 18| = 18?
Medium
Explanation6x + 18 = 18 or 6x + 18 = −18, which solve to x = 0 and x = -6.
68
What are the solutions to |x| + 15 = 19?
Medium
Explanation|x| + 15 = 19 means |x| = 4, so x = 4 or x = -4.
69
How many solutions does the equation |x + 40| = 15 have?
Medium
Explanation|x + 40| = 15: since 15 is positive, there are two solutions.
70
Which of the following numbers lies between -43 and -25?
Medium
Explanation-39 lies strictly between -43 and -25, while the other options fall outside that range.
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Explanation|-77 ÷ 11| = |-77| ÷ |11| = 77 ÷ 11 = 7.
72
What are the solutions to |x| + 18 = 21?
Medium
Explanation|x| + 18 = 21 means |x| = 3, so x = 3 or x = -3.
73
Which is farther from 1 on the number line: 10 or -77?
Medium
ExplanationDistance from 1: |10 − 1| = 9, and |-77 − 1| = 78. So -77 is farther.
Explanation|-2 ÷ 1| = |-2| ÷ |1| = 2 ÷ 1 = 2.
75
How many solutions does |x + 20| = −26 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x + 20| = −26 has no solution.
76
Which is greater: |1| or |-15|?
Medium
Explanation|1| = 1 and |-15| = 15, so |-15| is greater.
Explanation|10 × 10| = |10| × |10| = 10 × 10 = 100.
Explanation|41| − |-24| = 41 − 24 = 17. (Note this is generally different from |41 − (-24)|.)
Explanation|-15| − |43| = 15 − 43 = -28. (Note this is generally different from |-15 − 43|.)
80
What is the midpoint between -33 and -25 on the number line?
Medium
ExplanationThe midpoint of -33 and -25 is (-33 + -25) / 2 = -58/2 = -29.
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81
If |x| = |-35|, what are the possible values of x?
Medium
Explanation|x| = |-35| = 35 means x is 35 units from 0, so x = 35 or x = -35.
82
For a = 17 and b = 19, is |a + b| equal to |a| + |b|, or strictly less than |a| + |b|?
Medium
Explanation|a + b| = 36 and |a| + |b| = 36. Since a and b have the same sign, the two are equal.
83
For a = 8 and b = -9, is |a + b| equal to |a| + |b|, or strictly less than |a| + |b|?
Medium
Explanation|a + b| = 1 and |a| + |b| = 17. Since a and b have opposite signs, the two are not equal, with |a+b| smaller.
84
What are the solutions to |x + 32| = 6?
Medium
Explanation|x + 32| = 6 means x is 6 away from -32, so x = -32 + 6 = -26 or x = -32 − 6 = -38.
85
Which is farther from -29 on the number line: -15 or -48?
Medium
ExplanationDistance from -29: |-15 − (-29)| = 14, and |-48 − (-29)| = 19. So -48 is farther.
86
Which is farther from 10 on the number line: 80 or 35?
Medium
ExplanationDistance from 10: |80 − 10| = 70, and |35 − 10| = 25. So 80 is farther.
87
What are the solutions to |x − 29| = 28?
Medium
Explanation|x − 29| = 28 means x is 28 away from 29, so x = 29 + 28 = 57 or x = 29 − 28 = 1.
88
How many solutions does |x + 20| = −27 have?
Medium
ExplanationAn absolute value can never equal a negative number, so |x + 20| = −27 has no solution.
89
What is the solution set of |x − 32| < 25?
Medium
Explanation|x − 32| < 25 means x is within 25 of 32, giving 7 < x < 57.
90
How many integers satisfy |x + 24| < 3?
Medium
Explanation|x + 24| < 3 means -27 < x < -21. The integers strictly between -27 and -21 number 5.
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91
What is the solution set of |x + 39| > 15?
Medium
Explanation|x + 39| > 15 means x is farther than 15 from -39, giving x < -54 or x > -24.
92
How many integers satisfy |x − 28| < 4?
Medium
Explanation|x − 28| < 4 means 24 < x < 32. The integers strictly between 24 and 32 number 7.
Explanation|90 ÷ -15| = |90| ÷ |-15| = 90 ÷ 15 = 6.
94
What is the solution set of |x − 40| > 23?
Medium
Explanation|x − 40| > 23 means x is farther than 23 from 40, giving x < 17 or x > 63.
95
What is the solution set of |x + 39| < 12?
Medium
Explanation|x + 39| < 12 means x is within 12 of -39, giving -51 < x < -27.
96
What is the midpoint between -56 and 19 on the number line?
Medium
ExplanationThe midpoint of -56 and 19 is (-56 + 19) / 2 = -37/2 = -37/2.
97
Which of the following numbers lies between -39 and 0?
Medium
Explanation-36 lies strictly between -39 and 0, while the other options fall outside that range.
98
Which of the following numbers lies between -41 and -12?
Medium
Explanation-31 lies strictly between -41 and -12, while the other options fall outside that range.
99
How many integers satisfy |x + 14| < 15?
Medium
Explanation|x + 14| < 15 means -29 < x < 1. The integers strictly between -29 and 1 number 29.
100
What is the length of the solution interval for |x − 40| < 4?
Medium
Explanation|x − 40| < 4 describes the interval (36, 44), which has length 44 − (36) = 8.