Inequalities
1. Core Idea
An inequality is solved exactly like an equation, with one extra rule:
Multiplying or dividing both sides by a negative number reverses the inequality sign.
Because of that rule, and because inequalities describe ranges rather than points, they are the natural home of quantitative comparison questions.
2. Must-Know Rules
| Operation | Effect on the sign |
|---|---|
| Add or subtract anything | unchanged |
| Multiply/divide by a positive | unchanged |
| Multiply/divide by a negative | reversed |
| Take reciprocals of two positives | reversed (2 < 3 but 1/2 > 1/3) |
| Square both sides | only safe if both sides are non-negative |
Compound inequalities:
a < x < bmeans BOTH conditions — an interval.x < a or x > bmeans EITHER — two rays.
Operate on all three parts of a compound inequality at once: from -3 < 2x + 1 < 7, subtract 1 throughout, then divide by 2, giving -2 < x < 3.
Adding and multiplying inequalities:
- You may add two inequalities pointing the same way: if a > b and c > d, then a + c > b + d.
- You may not subtract them. If a > b and c > d, nothing follows about a - c and b - d.
- You may multiply only if all quantities are positive.
Quadratic inequalities. Solve x^2 - 5x + 6 > 0 by factoring: (x-2)(x-3) > 0. The critical points are 2 and 3. Test each region:
- x < 2: (neg)(neg) = positive -> satisfies
- 2 < x < 3: (pos)(neg) = negative -> fails
- x > 3: (pos)(pos) = positive -> satisfies
Answer: x < 2 or x > 3.
For "< 0" the answer would be the middle region, 2 < x < 3.
3. Worked Examples
Example 1 — The sign flip
Solve -3x + 7 > 22.
-3x > 15
Divide by -3 and flip: x < -5.
Check with x = -6: -3(-6) + 7 = 25 > 22. Correct.
Example 2 — Combining ranges
If 2 < x < 5 and 1 < y < 3, what is the range of x - y?
For the maximum of x - y, take x as large and y as small as possible: just under 5 minus just over 1 -> just under 4.
For the minimum, take x small and y large: just over 2 minus just under 3 -> just over -1.
-1 < x - y < 4.
Note: you flip the y-range and add. This is the correct way to "subtract" inequalities.
Example 3 — Quadratic inequality
Solve x^2 <= 9.
x^2 - 9 <= 0 -> (x - 3)(x + 3) <= 0.
Critical points -3 and 3; the product is negative between them.
-3 <= x <= 3.
Common wrong answer: x <= 3 only, which forgets all the negative values.
4. GRE Traps
- Forgetting to flip when dividing by a negative. The most common inequality error, full stop.
- Dividing by a variable of unknown sign. From ax > a you cannot divide by a — you don't know its sign, or whether it is zero.
- Subtracting inequalities. Flip the second and add instead.
- x^2 < 9 giving x < 3. It gives -3 < x < 3.
- Squaring both sides carelessly. -5 < 2 is true, but 25 < 4 is false.
- Reciprocals across zero. 1/x > 1/y does not imply y > x when the signs differ.
- Treating an inequality's endpoints as included when the sign is strict. Watch < versus <=.
5. Speed Tricks
- For range questions, test the four corner combinations (min-min, min-max, max-min, max-max) and take the extremes. It works for +, -, x and /.
- Draw the number line. Shade the solution; it prevents AND/OR mix-ups.
- On quantitative comparison, hunt for a counterexample rather than trying to prove equality. Two disagreeing test values immediately give answer D.
- Keep everything positive by moving terms rather than multiplying by negatives — it sidesteps the flip rule entirely.
- Critical-point method works for any factored inequality: mark the roots, then test one value in each region.
6. Self-Check
Q1. Solve 5 - 2x <= 11.
Q2. If -4 < a < 2 and 1 < b < 6, what is the range of ab?
Q3. Solve x^2 - 4x < 0.
Answers
A1. -2x <= 6 -> x >= -3 (sign flipped).
A2. Test the four corners: (-4)(1) = -4, (-4)(6) = -24, (2)(1) = 2, (2)(6) = 12. Range is -24 < ab < 12.
A3. x(x - 4) < 0; critical points 0 and 4; negative between them: 0 < x < 4.
7. One-Line Summary for the Board
Divide by a negative, flip the sign. x^2 < k gives a two-sided range. Never subtract inequalities.