GMAT Inequalities From Scratch

Inequalities behave like equations most of the time, and the exceptions are where the marks are won and lost. Three rules cover almost every mistake people make: when the sign flips, what you can add, and what a squared term hides.

Inclusive or exclusive

x > 5 excludes 5. x ≥ 5 includes it. The extra bar under the symbol is the difference between a boundary you can land on and one you cannot.

The same applies to ranges. 3 < x < 5 means x sits strictly between them. 3 ≤ x ≤ 5 allows both endpoints.

Watch for mixed ranges like a < x ≤ b, where one end is inclusive and the other is not. These appear on the GMAT precisely because they are easy to skim past.

One thing to hold onto: unless a question says otherwise, x is not restricted to whole numbers. x > 5 allows 5.1.

The sign flips only for negative multiplication and division

Multiply or divide both sides by a positive number and nothing changes. x > 3 becomes 2x > 6.

Multiply or divide by a negative number and the direction reverses. x > 3 becomes −x < −3. Check it: if x is 4, then −4 is smaller than −3, not larger.

What matters is the sign of the thing you multiply by, not the signs already in the expression. Dividing −3x > 9 by positive 3 keeps the direction, even though there is a minus sign in the expression. Dividing the same thing by −3 flips it.

Adding and subtracting never flip

Add or subtract the same quantity from both sides and the direction is preserved, including when that quantity is negative.

People confuse this with the rule above, because "negative" appears in both. Subtracting 5 is not multiplying by a negative. Different operation, different rule, no flip.

You can add two inequalities, never subtract them

If a < b and c < d, then a + c < b + d. This works only when both point the same direction. If one is reversed, flip it first so the two smaller sides are together.

Subtraction is not allowed. When you want to subtract, multiply one inequality by −1 first, remembering to flip its direction, then add.

One detail on symbols: if you add a strict inequality to an inclusive one, the result is strict. Equality would require both to reach their boundaries, and one of them cannot.

The chain rule

If a < b and b < c, then a < c. Chain as many as you like, as long as every link points the same way. You cannot mix a < with a > in one chain.

Square roots need positives

You can take the square root of both sides when both sides are positive, since negatives have no square root.

The related trap runs the other direction. x² < 9 does not mean x < 3. Because x could be negative, the correct answer is −3 < x < 3. Test it: x = −1 gives 1, which is under 9, so −1 must be in the range. x = −4 gives 16, which is not, so the range has to stop at −3.

Any time you meet a squared term inside an inequality, ask what happens if the variable is negative.