Linear inequalities

Official Digital SAT skill — Algebra domain.

What this question tests

Linear inequalities test whether you can decide which numbers or points make an inequality true, not just solve an equation. On the Digital SAT, this skill often appears in two forms: a one-variable inequality where you solve for a range of values, or a two-variable inequality where you determine whether a point lies in the solution region relative to a boundary line. In both cases, the core idea is substitution and logical comparison: plug in a candidate and check whether the inequality sign is satisfied. For two-variable inequalities, you also need to understand the graph meaning, where the boundary line $y=mx+b$ separates the plane into two regions and the inequality tells you which side is allowed. This is tested because it blends algebraic manipulation with interpretation, and it rewards careful attention to signs and to whether boundary points are included.

What to know

A linear inequality compares two linear expressions using $<$, $\le$, $>$, or $\ge$, and its solution is a set of values (a range on a number line or a region in the plane) rather than a single value.
To test a candidate value in one variable, substitute it into both sides and check whether the resulting statement, such as $3<7$ or $5\ge 5$, is true.
When solving a one-variable inequality, you can add or subtract the same quantity from both sides and multiply or divide both sides by a positive number without changing the inequality direction.
If you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign, for example from $<$ to $>$ or from $\ge$ to $\le$.
For two-variable inequalities, the boundary is the related line (often written as $y=mx+b$), and the inequality tells you whether solutions are above or below that line; strict inequalities ($<$ or $>$) exclude points on the boundary, while non-strict inequalities ($\le$ or $\ge$) include them.
To check whether a point $(x,y)$ satisfies an inequality like $y<mx+b$, substitute its coordinates and verify the comparison, for example confirm whether the given $y$ is less than the computed right-hand side.

How to approach it

First, identify whether the problem is asking about one variable on a number line or a point $(x,y)$ in the plane, because that determines whether you should solve for a range or test a coordinate pair.
If you are given answer choices that are specific values or points, use substitution as a fast and reliable method, because it directly checks the definition of a solution.
For a one-variable inequality, simplify carefully and keep track of operations, because each step must preserve the truth of the inequality and a sign mistake changes the solution set.
If you multiply or divide by a negative while isolating the variable, reverse the inequality sign, because negatives flip the order of numbers on the number line.
For a two-variable inequality, rewrite the boundary as $y=mx+b$ when possible, because then the comparison becomes a clear “is $y$ less than, equal to, or greater than the line’s $y$ value?” check.
When testing a point for a strict inequality, confirm it is on the correct side of the boundary and not on the boundary itself, because equality fails for $<$ and $>$ even if the point lies exactly on the line.
If multiple choices seem plausible, compute the right-hand side exactly and compare numerically, because estimation can be misleading when values are close or signs differ.

Common traps

Sign-flip slip: students forget to reverse the inequality after dividing by a negative, because they treat it like an equation step; avoid it by saying “negative division flips the sign” out loud as you do it.
Boundary confusion: students include a point on the line for $<$ or $>$, because they see a correct-looking value; avoid it by checking whether the symbol is strict ($<$, $>$) or inclusive ($\le$, $\ge$).
Direction mistake with $y<mx+b$: students pick a point with a $y$ that is actually above the line, because they compare the wrong quantity; avoid it by computing $mx+b$ first and then comparing the given $y$ to that number.
Arithmetic misread: students substitute correctly but miscalculate $mx+b$, because of sign errors like $-4(-4)$; avoid it by writing intermediate results and checking multiplication signs.
Over-solving when choices exist: students fully solve the inequality when a quick substitution would work, because they default to routine; avoid it by testing choices directly when that is faster and less error-prone.

Tips & shortcuts

If choices are points, compute the boundary value $mx+b$ once per choice and compare, and stop as soon as you find a true inequality if only one answer is correct.
Use a quick sign check before finalizing: negative times negative is positive, and that single fact prevents many wrong comparisons.
For strict inequalities, treat equality as an automatic rejection, which is a fast way to eliminate tempting boundary points.
If you solve a one-variable inequality, pick a test value from your solution set and one from outside it to sanity-check the direction, which catches sign-flip errors.

Worked example

Which value satisfies the inequality $-3(2x - 5) \ge 4x + 7$?

A. $1.0$
B. $0.9$
C. $1.4$
D. $0.6$ ✓ (correct answer)

Why: Distribute on the left: $-3(2x - 5) = -6x + 15$. So the inequality is $-6x + 15 \ge 4x + 7$. Subtract $7$ from both sides: $-6x + 8 \ge 4x$. Subtract $4x$ from both sides: $8 \ge 10x$. Divide both sides by $10$ (positive, so the inequality direction stays the same): $0.8 \ge x$, which is the same as $x \le 0.8$. Only $0.6$ satisfies $x \le 0.8$, so the correct answer is choice C.

Use the Practice Questions for this skill to drill it, then attempt a Timed Practice Test.