GRE Quantitative Quiz: Absolute Value Order

12 questions · exam conditions

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Absolute Value OrderQuestion 1 of 12

Which of the following statements is true for all real numbers $x$ ?

|x-2|=x-2

|x|\ge x

|x|\le x

|x+1|=|x|+1

|x|=-x

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GRE Quantitative Quiz

GRE Quantitative Quiz: Absolute Value Order

Practice Absolute Value Order in GRE Quantitative with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Absolute Value Order, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE Quantitative.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Which of the following statements is true for all real numbers $x$ ?

$|x-2|=x-2$
$|x|\ge x$ (correct answer)
$|x|\le x$
$|x+1|=|x|+1$
$|x|=-x$

Explanation: This question tests properties of absolute values that hold for all real numbers. The absolute value |x| represents the distance from x to zero, which is always non-negative. For any real number x, we have |x| ≥ 0. When x ≥ 0, |x| = x, so |x| ≥ x with equality. When x < 0, |x| = -x > 0 > x, so |x| > x. Therefore, |x| ≥ x is always true. The incorrect option |x| ≤ x would require |x| = x for all x, which fails when x is negative since then |x| = -x > x.

Question 2

If $x$ and $y$ are real numbers such that $|x-2|=|y-2|$ and $x<2<y$ , which of the following statements is true?

$x+y=4$ (correct answer)
$xy=4$
$x=y$
$x+y=2$
$x-y=4$

Explanation: This question tests the geometric interpretation of absolute value equations. The equation |x-2| = |y-2| means that x and y are equidistant from 2 on the number line. Since we're told x < 2 < y, x must be to the left of 2 and y to the right of 2. For these points to be equidistant from 2, we need 2-x = y-2, which gives us x + y = 4. This makes geometric sense: if x is d units to the left of 2 and y is d units to the right of 2, then x = 2-d and y = 2+d, so x + y = 4. A common error would be to think x and y are equal, but they're actually reflections of each other across the point 2.

Question 3

If $a$ and $b$ are real numbers and $|a|<|b|$ with $b<0$ , which of the following statements is true?

$a<b$
$a>b$
$a$ must be negative
$a$ could be either positive or negative (correct answer)
$a=b$

Explanation: This question tests the relationship between absolute values and ordering. The absolute value |a| represents the distance from a to zero, regardless of sign. Given that |a| < |b| and b < 0, we know b is negative and its distance from zero is greater than a's distance from zero. Since |b| = -b (because b is negative), we have |a| < -b. This means a could be any value between b and -b, which includes both positive and negative values. For example, if b = -10, then |b| = 10, and a could be any value with |a| < 10, such as a = 5 or a = -7. The key insight is that knowing |a| < |b| doesn't determine the sign of a.

Question 4

Real numbers $a$ and $b$ satisfy $a<0<b$ and $|a|>|b|$ . Which of the following statements is true?

$a+b>0$
$a+b<0$ (correct answer)
$|a+b|=|a|+|b|$
$|a|<|b|$
$a>b$

Explanation: This question tests absolute value relationships and ordering. The absolute value |a| represents the distance from a to zero, and similarly for |b|. Given a < 0 < b, we know a is negative and b is positive. Since |a| > |b|, the distance from a to zero exceeds the distance from b to zero. This means the negative value a has greater magnitude than the positive value b, so a + b < 0. For example, if a = -5 and b = 3, then |a| = 5 > |b| = 3, and a + b = -2 < 0.

Question 5

Let $t$ be a real number such that $|t-4|\ge 3$ . Which of the following values satisfies the inequality?

$2$
$6$
$5$
$1$ (correct answer)
$4$

Explanation: This question tests understanding of absolute value inequalities. The inequality |t-4| ≥ 3 means the distance from t to 4 is at least 3 units. This occurs when t ≤ 1 or t ≥ 7, representing points that are 3 or more units away from 4 on the number line. Among the given choices, we need to check which values satisfy this condition: |2-4| = 2 < 3 (no), |6-4| = 2 < 3 (no), |5-4| = 1 < 3 (no), |1-4| = 3 ≥ 3 (yes), |4-4| = 0 < 3 (no). Only t = 1 satisfies the inequality. A common mistake is to think values close to 4 satisfy the inequality, when actually we need values far from 4.

Question 6

If $x$ is a real number such that $|x-5|<|x-1|$ , which of the following statements is true?

$x<3$
$x=3$
$x>3$ (correct answer)
$x\le 3$
$x\ge 3$

Explanation: This question tests comparing distances using absolute value inequalities. The inequality |x-5| < |x-1| means x is closer to 5 than to 1 on the number line. To find when this occurs, we can square both sides (since both are non-negative): (x-5)² < (x-1)². Expanding gives x²-10x+25 < x²-2x+1, which simplifies to -10x+25 < -2x+1, then -8x < -24, so x > 3. Geometrically, this makes sense: the point x = 3 is equidistant from 1 and 5, and any point to the right of 3 is closer to 5. A common error is to try case analysis without recognizing the geometric interpretation of being closer to one point than another.

Question 7

If $x$ is a real number and $-2<x<1$ , which of the following is greatest?

$x$
$|x|$
$-x$
$|x+1|$
$|x-1|$ (correct answer)

Explanation: This question tests finding maximum values given a range constraint. With -2 < x < 1, we need to evaluate each expression across this interval. For x itself, the range is (-2, 1) with maximum approaching 1. For |x|, the maximum is 2 when x approaches -2. For -x, the range is (-1, 2) with maximum approaching 2. For |x+1|, since x+1 ranges from (-1, 2), the maximum is 2. For |x-1|, since x-1 ranges from (-3, 0), we have |x-1| ranging from 0 to 3, with maximum 3 when x approaches -2. Therefore, |x-1| achieves the greatest value of 3. A common error is to evaluate only at the endpoints without considering how absolute value affects negative inputs.

Question 8

Let $p$ and $q$ be real numbers such that $p<q$ and $|p|=|q|$ . Which of the following statements is true?

$p=q$
$p=-q$ (correct answer)
$p$ and $q$ are both positive
$p$ and $q$ are both negative
$p+q<0$

Explanation: This question tests the relationship between absolute value and ordering. The condition |p| = |q| means p and q have the same distance from zero. Combined with p < q, there are only two possibilities: either both are positive (impossible since then p < q would contradict |p| = |q|), or p is negative and q is positive with p = -q. Since p < q and |p| = |q|, we must have p < 0 < q with p = -q. This means p and q are opposites, confirming that p = -q. A common mistake is to think p and q could both be negative, but if they were both negative with equal absolute values, they would be equal, contradicting p < q.

Question 9

Let $a$ and $b$ be real numbers such that $a<b$ and $|a|=|b|$ . Which of the following statements is true?

$a+b>0$
$a=b$
$a=-b$ (correct answer)
$ab>0$
$|a-b|=0$

Explanation: This question tests the relationship between absolute value and ordering. The absolute value |a| represents the distance from a to zero, and similarly for |b|. Given |a| = |b|, both numbers are equidistant from zero. Combined with a < b, this means a must be negative and b must be positive, with a = -b. For example, if a = -3 and b = 3, then |a| = |b| = 3 and a < b. This confirms that a = -b is the true statement.

Question 10

If $|x|<|y|$ and $x<0<y$ , which of the following statements is true?

$x+y<0$
$x+y=0$
$x+y>0$ (correct answer)
$x>y$
$|x+y|=|y|-|x|$

Explanation: This question tests relationships between absolute values and signs of numbers. The absolute value |x| represents the distance from x to zero, regardless of sign. Given |x| < |y| and x < 0 < y, we know x is negative and y is positive. Since |x| < |y|, the distance from x to zero is less than the distance from y to zero. This means -x < y (since |x| = -x when x < 0). Adding x to both sides gives 0 < x+y, so x+y > 0. The incorrect option x+y = 0 would require |x| = |y|, contradicting our given condition.

Question 11

Real numbers $x$ and $y$ satisfy $|x|=|y|$ and $x<y$ . Which of the following statements is true?

$x=y$
$x$ and $y$ are both positive
$x+y=0$ (correct answer)
$x$ and $y$ are both negative
$x+y>0$

Explanation: This question tests the relationship between absolute values and ordering. The absolute value |x| represents the distance from x to zero, and |x| = |y| means x and y are equidistant from zero. Given x < y, there are two cases: both positive (impossible since then x = y), or x negative and y positive. In the second case, -x = y (since |x| = -x when x < 0 and |y| = y when y > 0). This gives us x + y = x + (-x) = 0. The incorrect option x = y contradicts the given constraint x < y.

Question 12

Real numbers $x$ and $y$ satisfy $|x-2|=|y-2|$ and $x<2<y$ . Which of the following statements is true?

$x+y=4$ (correct answer)
$xy=4$
$x=y$
$x+y=2$
$xy=2$

Explanation: This question tests understanding of absolute value equations and ordering. The absolute value |x-2| represents the distance from x to 2 on the number line, and similarly |y-2| represents the distance from y to 2. Since |x-2| = |y-2|, both x and y are equidistant from 2. Given that x < 2 < y, we know x is to the left of 2 and y is to the right of 2. If they're equidistant from 2, then 2-x = y-2, which simplifies to x+y = 4. The tempting incorrect option x = y fails because it ignores the constraint that x and y are on opposite sides of 2.