Linear inequalities look almost identical to linear equations, with one rule that changes everything: dividing or multiplying by a negative number flips the inequality sign. This guide covers one-variable and two-variable inequalities, compound inequalities, graphing shaded regions, and where students most often go wrong.
Solving one-variable inequalities
A linear inequality in one variable, like \(3x - 5 < 16\) or \(x \le 5\), is solved exactly like an equation — simplify, isolate the variable — except for one rule: whenever you multiply or divide both sides by a negative number, the direction of the inequality flips.
- \(x < 5\) means every number strictly less than 5 is a solution.
- \(x \le 5\) means 5 itself is included as a solution.
- The solution set is a range of infinitely many numbers, not a single value.
Worked example: one-variable inequality
Solve: \(-2(x - 3) \ge 10\)
- Distribute: \(-2x + 6 \ge 10\)
- Subtract \(6\) from both sides: \(-2x \ge 4\)
- Divide both sides by \(-2\), flipping the inequality: \(x \le -2\)
Skipping the flip in step 3 is the single most common error on this topic — it's worth pausing every time you divide by a negative number to double-check the sign flipped.
Compound inequalities: two inequalities at once
A compound inequality, such as \(-3 < 2x - 1 \le 5\), squeezes the variable between two bounds simultaneously and reads as two conditions joined together. The fastest way to solve it is to perform the same operation on all three parts at once, rather than splitting it into two separate inequalities first:
- Add \(1\) to all three parts: \(-2 < 2x \le 6\)
- Divide all three parts by \(2\): \(-1 < x \le 3\)
The solution is every value of \(x\) greater than \(-1\) and less than or equal to \(3\). The same negative-flip rule applies here too: if you ever multiply or divide all three parts by a negative number, all three inequality symbols flip direction, and the smaller and larger bounds effectively swap places in the written inequality.
Two-variable inequalities and graphing regions
A linear inequality in two variables, such as \(y > 2x - 4\), doesn't describe a line — it describes an entire shaded region of the coordinate plane. To graph one:
- Graph the boundary line as if the inequality were an equation (\(y = 2x - 4\)).
- Use a solid line if the inequality includes equality (\(\le\) or \(\ge\)); use a dashed line if it's strict (\(<\) or \(>\)).
- Shade above the line if the inequality is "greater than," or below it if "less than."
- Check a test point, like \((0,0)\), by substituting into the inequality — if it's true, you shaded the correct side.
Worked example: system of two inequalities
Which region satisfies both \(y \le x + 2\) and \(y > -x + 1\)?
- Graph \(y = x + 2\) as a solid line (since \(\le\) includes equality) and shade below it.
- Graph \(y = -x + 1\) as a dashed line (since \(>\) is strict) and shade above it.
- Test the point \((0,0)\) in each: for the first, \(0 \le 2\) is true; for the second, \(0 > 1\) is false — so \((0,0)\) is not in the overlap.
- The solution is the overlapping region above the dashed line and below (or on) the solid line — every point satisfying both conditions at once.
This mirrors solving a system of two linear equations exactly, except the answer is a region instead of a single point.
Worked example: an inequality word problem
A shipping service charges \(\$5\) per package plus a one-time \(\$20\) account fee. A customer has budgeted no more than \(\$95\) for shipping this month. What is the greatest number of packages they can ship?
- Let \(p\) represent the number of packages shipped.
- "No more than \(\$95\)" translates to \(\le 95\), giving the inequality \(20 + 5p \le 95\).
- Subtract \(20\) from both sides: \(5p \le 75\)
- Divide both sides by \(5\) (positive, so no flip needed): \(p \le 15\)
The greatest whole number of packages satisfying \(p \le 15\) is \(15\) itself. Notice the phrase "no more than" translates directly to \(\le\), not \(<\) — a distinction that matters just as much in the setup as the negative-flip rule matters in the solving.
Common inequality mistakes
| Mistake | Fix |
|---|---|
| Forgetting to flip the sign when dividing by a negative | Pause every time you divide/multiply by a negative number and flip automatically |
| Using a solid line for a strict inequality | Match the line style to the symbol: \(\le, \ge\) = solid; \(<, >\) = dashed |
| Shading the wrong side of the line | Always test a point not on the line, like \((0,0)\), to confirm |
| Misreading "at least" or "at most" in a word problem | "At least" means \(\ge\); "at most" or "no more than" means \(\le\) — translate before solving |
This is one of several patterns covered more broadly in The Most Common SAT Algebra Mistakes (and How to Avoid Them), which is worth a full read if inequalities are consistently where you lose points.
Tip: on grid-in questions asking for "the least value" or "the greatest value" satisfying an inequality, solve normally and pay close attention to whether the boundary itself (\(\le\)/\(\ge\)) is included.
Frequently asked questions
Why does dividing by a negative number flip the inequality?
Because negative numbers reverse order: if \(2 < 3\), multiplying both sides by \(-1\) gives \(-2\) and \(-3\), and \(-2 > -3\), not \(-2 < -3\). The flip keeps the statement true.
Do grid-in questions ever ask about inequalities?
Grid-in (student-produced response) questions typically ask for a specific boundary value or a specific point within a solution region, rather than the entire inequality — so you still need to solve the inequality fully before picking the requested value. Since a grid-in box can only hold one number, the question is always designed to have a single correct value to enter, such as "the greatest possible integer value of \(x\)," even though the inequality itself describes a whole range.
How is a compound inequality different from a system of two inequalities?
A compound inequality squeezes one variable between two bounds in a single line, like \(-1 < x \le 3\), and its solution is a range on a number line. A system of two inequalities involves two variables and two separate inequality statements, and its solution is a two-dimensional shaded region on the coordinate plane — a different shape of answer entirely, even though both use the same flip rule.
How should I practice this topic?
Work through the Linear inequalities set on topic-wise SAT Math practice at Easy, Medium, and Hard difficulty. The built-in Desmos graphing calculator is especially useful here — graph both the boundary line and test points to confirm your shading before committing to an answer.