GRE Quant · Study & Practice

Inequalities

AreaAlgebra DifficultyCore GRE weightageMedium–High — the natural language of Quantitative Comparison and every "range of values" question

An inequality is an equation with a wider mouth: instead of one exact value, it describes a whole range of values. You solve it almost exactly like an equation — same balancing moves — with one rule that changes everything: multiply or divide by a negative number and the inequality sign flips. That single rule is the most tested and most forgotten idea in the topic. Inequalities matter disproportionately on the GRE because Quantitative Comparison is itself an inequality question — you are always deciding whether one quantity is greater, less, or equal. This chapter covers solving linear inequalities, the sign-flip trap in full, compound inequalities that trap a variable between two bounds, and absolute-value inequalities, which split into two cases. Master these and you are reading the GRE's native language.

Topics

1 Solving Linear Inequalities 4 worked · 5 practice · 8-Q test Start → 2 Flipping the Sign 4 worked · 5 practice · 8-Q test Start → 3 Compound Inequalities 4 worked · 5 practice · 8-Q test Start → 4 Absolute-Value Inequalities 4 worked · 5 practice · 8-Q test Start →

⚡ GRE shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Solve inequalities exactly like equations, with ONE change: flip the sign whenever you multiply or divide by a negative.
  • To dodge the flip entirely, move the variable to the side that keeps its coefficient positive, then divide by a positive.
  • For all three parts of a compound inequality, apply each operation to every part at once.
  • |expression| < k ⇒ one band (−k to k); |expression| > k ⇒ two rays (< −k or > k). "Less" is AND, "greater" is OR.
  • If k is negative: |x| < k has no solution; |x| > k is true for all x. Check the sign of k before solving.
  • On Quantitative Comparison, the sign flip is often the whole game — track the direction of the inequality carefully.

⚠️ Common mistakes & traps

GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Forgetting to flip the inequality when multiplying or dividing by a negative number.
  • Flipping the sign when merely adding or subtracting a negative — the flip is only for multiply/divide.
  • Treating |x| > k as a single band instead of two separate regions (or |x| < k as two rays).
  • In a compound inequality, operating on only two of the three parts.
  • Not swapping the end bounds when a negative multiplier flips a compound inequality.
  • Missing the no-solution / all-real edge cases when the constant on an absolute-value inequality is negative.

📈 GRE exam insight & question patterns

Quantitative Comparison — if −2x > 8, Quantity A is x and Quantity B is −5. Which is greater?

Divide by −2 and flip: x < −4. Since x is below −4, it could be −4.5 (> −5) or −10 (< −5), so the answer is "cannot be determined".

Numeric Entry — what is the largest integer value of x satisfying 3x − 4 < 11?

3x < 15 ⇒ x < 5. The largest integer is 4 (strict inequality excludes 5).

A multiple-answer item asks which values satisfy |x − 2| < 3 from a list including 0, 4, 5, 6. Which apply?

The band is −1 < x < 5, so 0 and 4 qualify; 5 is excluded (strict) and 6 is out.

Why does |x| < −2 have no solution while |x| > −2 has infinitely many?

Absolute value is never negative, so it cannot be below −2 (no solution) but is always above −2 (all real x).

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

Multiply/divide an inequality by a negativeTap to reveal
flip the inequality sign
Add/subtract a negativeTap to reveal
sign does NOT flip
|x| < k (k > 0)Tap to reveal
−k < x < k (one band)
|x| > k (k > 0)Tap to reveal
x < −k or x > k (two rays)
|x − a| < kTap to reveal
a − k < x < a + k
|x| < k with k < 0Tap to reveal
no solution
|x| > k with k < 0Tap to reveal
all real numbers
Compound a < x < bTap to reveal
do each operation to all three parts

📌 Quick revision

An inequality describes a range of values and solves like an equation — with the one non-negotiable twist that multiplying or dividing by a negative flips the sign. Sidestep that trap by rearranging so the variable's coefficient stays positive. Compound inequalities pin a variable between two bounds; operate on all three parts together, and swap the ends if a negative multiplier flips them. Absolute-value inequalities split by direction: "less than" gives a single band (AND), "greater than" gives two rays (OR), and a negative constant creates the no-solution / all-real edge cases the GRE loves. Because Quantitative Comparison is itself an inequality question, this chapter is really training in the exam's native language — the direction of the sign is often the entire answer.

Chapter test

📝 Inequalities — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara GRE success checklist

You have truly mastered Inequalities when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall8 cards