How to solve SAT linear inequalities questions

Linear inequalities work like linear equations, except the answer is a range of values instead of a single number. On the Digital SAT you solve one-variable inequalities, turn word problems about limits and budgets into inequalities, and decide which points satisfy an inequality or a system of inequalities in the $xy$-plane. Algebra is about 35% of the Math section, or 13 to 15 of the 44 questions, and a few of those, often 2 or 3, test this skill.

What the questions look like

Solve or test a one-variable inequality. You solve something like $-3x + 7 > 19$, or pick which value of $x$ is a solution. Compound inequalities such as $-2 < 2x + 4 \leq 10$ also appear.

Model a limit. A situation has a maximum or minimum, like a weight limit, a budget, or a required score. You write the inequality or find the greatest or least whole number that works.

Two-variable inequalities and graphs. You choose which point satisfies a system like $y > 2x - 3$ and $x + y \leq 6$, or which inequality matches a shaded region in a graph.

Step-by-step method

1. Translate the words.
   • "At least" and "no less than" mean $\geq$.
   • "At most" and "no more than" mean $\leq$.
   • "More than" means $>$, and "fewer than" or "less than" means $<$.
2. Solve like an equation, with one extra rule. When you multiply or divide both sides by a negative number, flip the inequality sign. For example, $-3x + 7 > 19$ becomes $-3x > 12$, and dividing by $-3$ gives $x < -4$.
3. For compound inequalities, do the same operation to all three parts at once. From $-2 < 2x + 4 \leq 10$, subtract 4 to get $-6 < 2x \leq 6$, then divide by 2 to get $-3 < x \leq 3$.
4. Round with the context in mind. If you need a whole number of items and the limit is a maximum, round down. If you need to reach a minimum, round up.
5. Test points for two-variable questions. Substitute the point into every inequality in the system. A point is a solution only if it satisfies all of them.
6. Read graphs carefully. A solid boundary line means $\leq$ or $\geq$. A dashed line means $<$ or $>$. When $y$ is isolated on the left, $y >$ shades above the line and $y <$ shades below.

Worked example

Example 1. An elevator has a maximum load of 1,600 pounds. A worker who weighs 190 pounds is riding the elevator with boxes that weigh 42 pounds each. What is the greatest number of boxes the worker can bring in one trip?

Let $b$ be the number of boxes. The worker plus the boxes must be at most 1,600 pounds:

$$190 + 42b \leq 1600$$
$$42b \leq 1410$$
$$b \leq 33.57\ldots$$

The number of boxes must be a whole number, and 34 would break the limit, so round down. The answer is 33.

Check both sides of the cutoff: $190 + 42(33) = 1576$, which is under 1,600. $190 + 42(34) = 1618$, which is over.

Example 2. Which point $(x, y)$ is a solution to the system below?

$$y > 2x - 3$$
$$x + y \leq 6$$

A) $(4, 4)$ B) $(1, 5)$ C) $(0, -4)$ D) $(3, 3)$

Test each point in both inequalities.

• $(4, 4)$: Is $4 > 2(4) - 3 = 5$? No. Eliminate it.
• $(1, 5)$: Is $5 > 2(1) - 3 = -1$? Yes. Is $1 + 5 \leq 6$? Yes, since $6 \leq 6$. This point works.
• $(0, -4)$: Is $-4 > -3$? No.
• $(3, 3)$: Is $3 > 2(3) - 3 = 3$? No, because 3 is not greater than 3.

The answer is B. Choice D sits exactly on the dashed boundary of the first inequality, so it is not included. Choice B sits on the solid boundary of the second, so it is included.

Common traps

• Forgetting to flip the sign. This happens most when the negative is hidden, as in $\frac{x}{-2} \geq 5$. Any time you multiply or divide by a negative, flip.
• Rounding the wrong way. In Example 1, rounding $33.57$ to 34 feels natural but breaks the weight limit. Ask whether the context sets a ceiling or a floor.
• Treating $<$ and $\leq$ the same at the boundary. A point on the boundary line satisfies $\leq$ but not $<$. Check boundary cases like Choice D carefully.
• Translating phrases backward. "No more than 20" means $\leq 20$. Students often see "more" and write $>$.
• Shading from an unsimplified inequality. For $-2y > 4x + 6$, dividing by $-2$ gives $y < -2x - 3$, which shades below the line. Isolate $y$ before deciding which side to shade.

Desmos tip

Type inequalities directly, like y > 2x - 3 and x + y <= 6. Desmos shades each region and draws strict inequalities with dashed lines. The overlap is the solution set. To test an answer choice, type the point, such as $(1, 5)$, and see whether it lands in the overlap. Zoom in for points near a boundary, since the picture alone can be misleading there.