Lesson 5 of 5 · Algebra
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Linear inequalities appear about 1–3 times per SAT. Most are word problems about budgets, limits and minimums, or questions about which points lie in a shaded region of the -plane. The algebra is the same as for equations, plus one extra rule. So most lost points come from translation and boundary details: flipping the sign, strict versus inclusive boundaries, and rounding whole-number answers.
| Question shape | Typical stem |
|---|---|
| Solve | “Which of the following describes all solutions to ?” |
| Integer answers | “What is the greatest integer value of that satisfies ...?” |
| Model | “Which inequality represents the possible numbers of packages?” |
3 questions on linear inequalities in one or two variables. Every option is explained once you answer.
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When must you reverse an inequality sign?
| Interpret | “Which of the following is the best interpretation of the solution in this context?” |
| Test points | “Which point is a solution to the system of inequalities?” |
| Regions | “Which system of inequalities is represented by the shaded region?” |
| Optimize | “What is the greatest number of adult tickets the group can buy?” |
Solve a linear inequality exactly like an equation, except that multiplying or dividing both sides by a negative number reverses the inequality:
You can avoid the flip by collecting the -terms on the side where the coefficient stays positive. For example, becomes , so , which is the same as .
To solve a compound inequality such as , apply each step to all three parts. Here that gives .
| Phrase | Symbol |
|---|---|
| at least, no less than, a minimum of, no fewer than | |
| at most, no more than, a maximum of, cannot exceed | |
| more than, greater than, exceeds | |
| less than, fewer than |
First build the expression for the quantity (total cost, total weight or total hours). Then attach the symbol and the limit. Units tell you which number goes with which expression: a dollar limit goes with the price expression, and an item limit goes with the count.
Contexts often require whole numbers (people, boxes, hours). Solve, then round in the direction the constraint allows:
Then check your whole number in the original inequality.
For an inequality solved for :
| Inequality | Boundary line | Shaded side |
|---|---|---|
| Dashed | Above | |
| Solid | Above | |
| Dashed | Below | |
| Solid | Below |
The solutions to a system of inequalities are the overlap of the shaded regions. A point is a solution only if it satisfies every inequality. When in doubt, test a point: substitute or any other convenient point that is not on a boundary.
Sometimes the two boundaries are parallel, as in and . When , the solutions form a strip between the lines. When , the two inclusive inequalities share only the line itself. When , there are no solutions at all. If either inequality is strict, also gives no solutions.
<= and >=), and it shades the overlap. Add the answer-choice points to see which one lands in the doubly shaded region.