Absolute Value — Practice Set

Bank code: NUM-ABS · Section: Arithmetic · 30 questions — Easy 5 · Medium 8 · Hard 12 · Extreme 5

|x| means distance from zero: equations split into two cases, less-than gives the inside of an interval and greater-than gives the two rays outside it.

Every question below is live in the question bank under the ID shown — the sheet and the portal are the same questions. Attempt a level with the clock running, then check Part C.

---

Part A — Questions

Quantitative Comparison — the four choices are always the same, so they are not reprinted: (A) Column A is greater · (B) Column B is greater · (C) The two quantities are equal · (D) The relationship cannot be determined from the information given

Level 1 · Easy — 5 questions · ~4 min

warm-up — these must be automatic

Q1 · NUM-ABS-003 · MCQ · 40s

If |x| = 9, what are all the possible values of x?

(A) x = 9 only
(B) x = -9 only
(C) x = 9 or x = -9
(D) x = 3 or x = -3
(E) There is no such x

Q2 · NUM-ABS-004 · Numeric Entry · 40s

What is the value of |3 - 8| ? Enter your answer as a number.

Numeric entry — write the number.

Q3 · NUM-ABS-006 · MCQ · 50s

On the number line, what is the distance between the points -4 and 7?

(A) -11
(B) 3
(C) 4
(D) 11
(E) Cannot be determined

Q4 · NUM-ABS-008 · MCQ · 55s

Which of the following is true for every real number x?

(A) |x| = x
(B) |x| = -x
(C) |x| > 0
(D) |x| >= 0
(E) |x| > x

Q5 · NUM-ABS-010 · QC · 45s

x is a nonzero real number.

Column A: |x| Column B: x

---

Level 2 · Medium — 8 questions · ~10 min

two or three steps, one planted trap each

Q6 · NUM-ABS-013 · MCQ · 75s

Which of the following describes all values of x that satisfy |2x + 1| > 5?

(A) -3 < x < 2
(B) x > 2
(C) x < -3
(D) x < -3 or x > 2
(E) x < -2.5 or x > 2.5

Q7 · NUM-ABS-014 · Numeric Entry · 65s

What is the product of all values of x that satisfy |x + 4| = 7? Enter your answer as a number.

Numeric entry — write the number.

Q8 · NUM-ABS-015 · QC · 75s

a and b are both negative.

Column A: |a + b| Column B: |a| + |b|

Q9 · NUM-ABS-016 · Select all that apply · 80s

Which of the following values of x satisfy |x - 1| <= 3? Select all that apply.

(A) -3
(B) -2
(C) 0
(D) 4
(E) 5

Q10 · NUM-ABS-017 · QC · 80s

x > 0

Column A: |x^2 - 4| Column B: x^2 - 4

Q11 · NUM-ABS-020 · Select all that apply · 90s

Which of the following are solutions to |2x - 4| = |x + 1|? Select all that apply.

(A) x = -5
(B) x = -1
(C) x = 1
(D) x = 5
(E) x = 7

Q12 · NUM-ABS-023 · Numeric Entry · 75s

How many integer values of x satisfy |x - 2| < 4? Enter your answer as a number.

Numeric entry — write the number.

Q13 · NUM-ABS-025 · MCQ · 80s

If |x + 2| = |x - 6|, what is the value of x?

(A) -4
(B) -2
(C) 2
(D) 4
(E) 6

---

Level 3 · Hard — 12 questions · ~20 min

where 162+ is won or lost

Q14 · NUM-ABS-027 · MCQ · 90s

How many integer values of x satisfy ||x| - 3| = 2?

(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

Q15 · NUM-ABS-028 · Numeric Entry · 105s

If |x - 3| + |x + 1| = 6, what is the greatest possible value of x? Enter your answer as a number.

Numeric entry — write the number.

Q16 · NUM-ABS-029 · MCQ · 95s

a and b are real numbers with |a - b| = 5 and |a + b| = 3. What is the value of a^2 - b^2?

(A) -15
(B) 0
(C) 15
(D) 16
(E) It cannot be determined from the information given

Q17 · NUM-ABS-030 · QC · 90s

x and y are nonzero real numbers.

Column A: |xy|/(xy) Column B: (|x|/x)(|y|/y)

Q18 · NUM-ABS-031 · Select all that apply · 95s

If |x^2 - 5| = 4, which of the following are possible values of x? Select all that apply.

(A) -3
(B) -1
(C) 2
(D) 3
(E) 9

Q19 · NUM-ABS-032 · MCQ · 100s

For how many integer values of x is |x^2 - 10x + 25| < 9?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

Q20 · NUM-ABS-033 · Numeric Entry · 105s

On the number line, point P is at -3 and point Q is at 9. Point R lies on the number line and its distance from P is twice its distance from Q. What is the sum of all possible coordinates of R? Enter your answer as a number.

Numeric entry — write the number.

Q21 · NUM-ABS-034 · QC · 90s

x and y are real numbers with xy < 0.

Column A: |x + y| Column B: |x - y|

Q22 · NUM-ABS-035 · MCQ · 100s

If |x - 5| < 2 and |y + 3| < 4, which of the following gives the complete set of possible values of x - y?

(A) -14 < x - y < 14
(B) -4 < x - y < 8
(C) 2 < x - y < 14
(D) 4 < x - y < 12
(E) 6 < x - y < 10

Q23 · NUM-ABS-036 · Select all that apply · 110s

Let f(x) = |x - 2| - |x + 2|. Which of the following statements are true? Select all that apply.

(A) f(0) = 0
(B) f(5) = 4
(C) f(-5) = -4
(D) The minimum value of f(x) is -4
(E) f(x) = 4 for every x > 2

Q24 · NUM-ABS-041 · Numeric Entry · 110s

What is the sum of all distinct real solutions to |2x - 6| = x^2 - 9? Enter your answer as a number.

Numeric entry — write the number.

Q25 · NUM-ABS-042 · QC · 90s

x < 3

Column A: sqrt(x^2 - 6x + 9) Column B: 3 - x

---

Level 4 · Extreme — 5 questions · ~10 min

165+ — expect to need the insight, not the grind

Q26 · NUM-ABS-037 · MCQ · 115s

For real numbers u and v, let d(u, v) = |u - v|. If d(x, 3) + d(x, 7) = d(3, 7), which of the following must be true?

(A) x = 5
(B) x = 3 or x = 7
(C) 3 <= x <= 7
(D) x < 3 or x > 7
(E) No real x satisfies the equation

Q27 · NUM-ABS-038 · Numeric Entry · 120s

What is the sum of all real solutions to |x^2 - 3x| = |x^2 - 3|? Enter your answer as a number.

Numeric entry — write the number.

Q28 · NUM-ABS-039 · QC · 110s

n is a positive integer.

Column A: |(-2)^n + (-2)^(n+1)| Column B: 2^n

Q29 · NUM-ABS-040 · Select all that apply · 125s

For which of the following values of k does the equation |x^2 - 4| = k have exactly two distinct real solutions? Select all that apply.

(A) k = -1
(B) k = 0
(C) k = 4
(D) k = 8
(E) k = 16

Q30 · NUM-ABS-043 · MCQ · 125s

The real numbers a, b and c satisfy |a| = 7, |b| = 4 and |c| = 2. What is the greatest possible value of |a + b| - |b + c|?

(A) 1
(B) 5
(C) 9
(D) 11
(E) 13

---