EQN-03-03

Inequalities

Solve inequalities like equations — with the one rule that flips the sign, the trap the GAT-E tests most.

~5 min read•Difficulty 2/5

Solve like an equation, with one twist

Use the same inverse operations as for equations. The only difference:

Multiplying or dividing both sides by a negative number reverses the inequality sign.

- 2 x < 6

x > - 3 (divide by - 2 \to flip < to >)

Adding or subtracting never flips the sign; only multiplying/dividing by a negative does.

Reading the solution

$x > - 3$ means every number to the right of $- 3$ on the number line. An inequality usually has infinitely many solutions, not one value.

Compound inequalities

- 1 \leq x < 4 means x is at least - 1 and less than 4.

Do the same operation to all three parts at once.

Sign-sensitive comparisons

If you don't know whether a variable is positive or negative, you cannot safely multiply an inequality by it — this is exactly why so many quantitative-comparison questions end in "cannot be determined".

Worked examples

Example 1 (difficulty 2/5)

Solve

: 3 - 2 x \geq 11

◎ Predict the answer first:

Example 2 (difficulty 2/5)

How many integers satisfy

- 2 < x \leq 3

◎ Predict the answer first:

Tips

Flip the inequality sign only when you multiply or divide by a negative.
Mark the boundary on a number line and shade the solution side to avoid direction errors.
Strict $(<, >)$ excludes the endpoint; non-strict $(\leq, \geq)$ includes it.

Common mistakes

Forgetting to flip the sign when dividing by a negative

Why: Inequalities feel just like equations.

Fix: Negative multiplier/divisor

\to

reverse the inequality direction.

Counting an excluded endpoint among integer solutions

Why: The two end symbols look alike.

Fix: Check each end

:< / >

excludes

, \leq / \geq

includes.

← Back to Algebra Practice this lesson →