Linear inequalities in one or two variables: solve and graph inequalities, remember to flip the sign when multiplying or dividing by a negative, and interpret feasible regions in context.

What this skill is asking

A linear inequality uses $<$, $\le$, $>$ or $\ge$ instead of $=$. The Digital SAT (Algebra domain) tests solving inequalities in one variable (whose solutions are ranges), graphing inequalities in two variables (whose solutions are half-planes), and translating real-world constraints such as budgets and capacities into inequalities. The algebra mirrors solving equations, with one extra rule.

The one rule that changes everything

Inequalities behave like equations until a negative multiplier appears.

A worked half-plane question

Two-variable inequalities are about shading the correct region.

One-variable inequalities and their ranges

Solving $-3x + 5 \ge 14$ proceeds exactly like an equation until the final division: subtract $5$ to get $-3x \ge 9$, then divide by $-3$ and flip to $x \le -3$. The solution is the range of all $x$ at most $-3$. A reliable check is to test a value: $x = -4$ gives $-3(-4) + 5 = 17 \ge 14$, true, confirming the direction. The sign-flip rule is the single most common source of errors on these questions, so build the habit of pausing whenever you divide or multiply by a negative.

Constraints and feasible regions

Word problems turn inequalities into constraints. A budget of \40 with items costing \3 and \$2 becomes$3n + 2p \le 40$; a requirement to make "at least 100 units" becomes a$\ge 100$constraint. Often several constraints apply at once (a budget, a minimum quantity, non-negative counts), and the set of points satisfying **all** of them is the **feasible region**. The SAT may ask whether a particular pair is feasible (test it in every inequality) or which inequality models a described limit (match "at most" to$\le$and "at least" to$\ge$, and build each term from its unit cost or rate).

A subtler version pairs an inequality with a system of inequalities and asks which point lies in the overlap, or for the maximum or minimum of some quantity over the feasible region. To test a point against a system, substitute it into every inequality; it is feasible only if it satisfies all of them. For a "greatest possible value" question, the answer typically sits at a corner of the feasible region (where two boundary lines meet), so finding those corner points and checking the quantity at each is the reliable route. Graphing the constraints in Desmos, where each inequality shades a half-plane and the feasible region is the overlap of the shadings, makes both the feasibility test and the corner search visual and quick.