QUANTITATIVE REASONING · GMAT-Q-INEQ

Linear Inequalities

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Model the relationship

a < b, c < 0 \Rightarrow a c > b c

Identify constraints and signs

L < x < U \Rightarrow x \in (L, U)

Verify with substitution

x < k, k \in Z \Rightarrow max (x \in Z) = k - 1

Think Like a GMAT Test-Taker

Solve the inequality while tracking whether each operation preserves or reverses its direction. Apply integer and strict-endpoint restrictions only after finding the real interval.

Faster Methods

Intersect the intervals from simultaneous conditions.

Check the candidate endpoint and its adjacent integer in the original inequality.

For a minimum feasible integer, test both your answer and the preceding integer. For a maximum, test the following integer.

Worked example 1

What is the least integer x satisfying 7−3x<−5?

- 3 x < - 12 \Rightarrow x > 4; x_{m i n} = 5

Division by negative three reverses the sign; four itself fails the strict inequality.

Worked example 2

How many integers satisfy −2<x≤4?

x \in {- 1, 0, 1, 2, 3, 4}; 6

Only the left endpoint is excluded. Count the integers, not the interval's length.

Worked example 3

If 2x+1≤9 and 5−x<3, what interval contains x?

x \leq 4, x > 2 \Rightarrow 2 < x \leq 4

Both conditions must hold, so take their intersection.

Worked example 4

A company has a fixed cost of $240, a variable cost of $8 per item, and revenue of $14 per item sold. What is the least whole number of items it must sell for a profit greater than $100?

14 n - (240 + 8 n) 6 n n > 100 > 340 > \frac{170}{3} \Rightarrow n_{m i n} = 57

Profit means revenue minus all costs. Solve the strict inequality and then apply the whole-number condition. At 56 items profit is $96, which fails; at 57 it is $102, which works. Rounding to the nearest integer is not a valid rule here.

Common Traps

Dividing by a negative reverses the inequality.
An endpoint is excluded by < or >, even if it is an integer.