Linear inequalities

Concept

Inequalities use the same algebra as equations, with one rule that catches everyone out:

Flip the inequality sign whenever you multiply or divide both sides by a negative number.
Adding or subtracting a number — even a negative one — does not flip the sign.

For graphing on a number line: an open circle means strict (< or >), a closed circle means inclusive (≤ or ≥).

For graphing on the x-y plane: draw the boundary line (dashed for strict, solid for inclusive), then shade the half-plane where the inequality is true. To pick the correct side quickly, test the origin (0, 0) — if it satisfies the inequality, shade the side containing the origin.

Word problems often combine several inequalities at once: a budget cap (≤), a minimum amount (≥), and non-negative quantities (x ≥ 0, y ≥ 0). The solution is the region where all inequalities are satisfied.

Worked example 1

Solve `−3x + 5 > 14` and graph the solution.

Solution

Step 1. Subtract 5 from both sides: −3x > 9

Step 2. Divide both sides by −3. Because we divided by a negative, flip the sign: x < −3

Solution: x < −3. On a number line: open circle at −3, arrow pointing left.

Worked example 2 · word problem

A florist has $50 to spend. Roses cost $4 each and tulips $2 each. The vase fits no more than 20 stems. Let r = roses and t = tulips. Write all the inequalities that describe a valid order.

Solution

Budget. Cost cannot exceed $50: 4r + 2t ≤ 50

Vase. Total stems no more than 20: r + t ≤ 20

Sense. Cannot buy a negative number of flowers: r ≥ 0 and t ≥ 0

A valid order is any (r, t) satisfying all four inequalities at once. The "feasible region" is the overlap of the four half-planes.

Practice test

8 questions on solving inequalities, graphing, and translating word problems into constraints.

Practice test · 8 questions Question 1 of 8 · Score 0