GRE Quantitative Flashcards: Absolute Value Order

What this deck covers

This deck focuses on Absolute Value Order, giving you a quick way to review the definitions, rules, and examples that matter most for GRE Quantitative.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify the correct order from least to greatest: $|-1|,\ -\frac{3}{2},\ \frac{4}{3}$.

Answer: $-\frac{3}{2}<|-1|<\frac{4}{3}$. Ordering starts with negative -1.5, then 1 from | -1 |, then positive 1.333.

Flashcard 2: What is the solution set of the inequality $|2x+1|\le 5$?

Answer: $-3\le x\le 2$. Solving |2x+1|<=5 gives -5 <= 2x+1 <=5, subtract 1 then divide by 2.

Flashcard 3: Which is larger: $-\frac{3}{2}$ or $\left| -\frac{5}{4} \right|$?

Answer: $\left| -\frac{5}{4} \right|=\frac{5}{4}$ is larger. Absolute value turns -5/4 positive to 1.25, which is greater than -1.5.

Flashcard 4: What is the smallest possible value of $|x|$ for real $x$, and when does it occur?

Answer: Minimum is $0$, occurring at $x=0$. The absolute value |x| is always non-negative, reaching its minimum of 0 only when x=0.

Flashcard 5: What is the solution set of the inequality $|x-3|<2$?

Answer: $1<x<5$. Solving |x-3|<2 expands to -2 < x-3 < 2, adding 3 to all parts.

Flashcard 6: Which is larger: $|2-9|$ or $|2-6|$?

Answer: $|2-9|=7$ is larger. Distance from 2 to 9 is 7, exceeding distance to 6 which is 4.

Flashcard 7: What inequality is equivalent to $|x|\le a$ for $a>0$?

Answer: $-a\le x\le a$. For positive a, |x|<=a includes all x within or at distance a from 0.

Flashcard 8: What is the definition of absolute value $|x|$ written as a piecewise function?

Answer: $|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}$. The absolute value function returns the non-negative value by taking x when x is non-negative and its negation when x is negative.

Flashcard 9: Which is larger: $|-7|$ or $|-5|$?

Answer: $|-7|$ is larger. Larger magnitude negative input to absolute value yields larger output, as 7>5.

Flashcard 10: What is the solution set of the inequality $|x+2|\ge 4$?

Answer: $x\le -6$ or $x\ge 2$. Solving |x+2|>=4 splits into x+2 <= -4 or x+2 >=4, subtracting 2.

Flashcard 11: What is the solution set of $|x|>0$?

Answer: $x\ne 0$. |x|>0 excludes only x=0 where |x|=0, including all other reals.

Flashcard 12: What is the solution set of $|x|<0$?

Answer: No solution; empty set. Since |x| is always >=0, it cannot be less than 0, yielding no solutions.

Flashcard 13: Which is larger: $|x-1|$ or $|x-4|$ when $x=0$?

Answer: $|0-4|$ is larger. At x=0, distance to 4 is 4, greater than distance to 1 which is 1.

Flashcard 14: What is the distance interpretation of $|a-b|$ on the number line?

Answer: $|a-b|$ is the distance between $a$ and $b$. On the number line, the absolute difference |a-b| measures the positive distance separating points a and b.

Flashcard 15: What inequality is equivalent to $|x|\ge a$ for $a>0$?

Answer: $x\le -a$ or $x\ge a$. For positive a, |x|>=a includes all x at or beyond distance a from 0.

Flashcard 16: Identify the correct order from least to greatest: $-2,\ | -3 |,\ 1$.

Answer: $-2<1<|-3|$. Ordering -2 (negative), then 1 (positive), then 3 from | -3 |.

Flashcard 17: What is the solution set of $|x|\ge a$ when $a<0$?

Answer: All real numbers. For negative a, |x|>=0 is always greater than or equal to a, so all reals satisfy.

Flashcard 18: What is the ordering rule comparing $|a|$ and $|b|$ using squares?

Answer: $|a|<|b|\iff a^2<b^2$. Squaring both sides preserves the inequality since squares equal the squares of absolutes and are monotonic for non-negative values.

Flashcard 19: What is the solution set of $|x|<a$ when $a<0$?

Answer: No solution; empty set. For negative a, |x|>=0 cannot be less than a negative number, so no solutions.

Flashcard 20: Which is larger: $-3$ or $|-4|$?

Answer: $|-4|=4$ is larger. Absolute value makes |-4| positive 4, which exceeds -3 on the number line.

Flashcard 21: What is the largest possible value of $|x|$ when $x$ ranges over all real numbers?

Answer: No largest value; $|x|$ is unbounded above. Since x can be arbitrarily large positive or negative, |x| can exceed any bound with no maximum.

Flashcard 22: What identity relates absolute value to a square root for real $x$?

Answer: $|x|=\sqrt{x^2}$. The square root of x squared yields the non-negative root, matching the absolute value.

Flashcard 23: What inequality is equivalent to $|x|>a$ for $a>0$?

Answer: $x<-a$ or $x>a$. For positive a, |x|>a means x is more than distance a from 0, either left of -a or right of a.

Flashcard 24: What is the solution set of $|x|\le 0$?

Answer: $x=0$. Since |x|>=0 and equals 0 only at x=0, |x|<=0 holds solely there.