10.2 Linear Inequalities
Key Takeaways
- Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol; adding or subtracting a negative does not.
- Inclusive inequalities (≤, ≥) include the boundary: a closed circle on a number line, or a solid line in the plane. Strict inequalities (<, >) use an open circle or a dashed line.
- For a two-variable inequality, graph the boundary line and test a point such as (0, 0) to decide which half-plane is the solution.
- In word problems, at least and no fewer than mean ≥; at most and no more than mean ≤; equality is included unless the sentence says more than or fewer than.
- A simple compound inequality such as 1 < x ≤ 5 is solved by performing the same operation on all three parts; check a number inside, on each end, and outside.
A linear inequality replaces the equals sign in a linear equation with <, >, ≤, or ≥. Solutions are usually a range, not a single number. College Board's Algebra list for PSAT 8/9 includes linear inequalities in one or two variables: create them from a constraint, solve them, and decide whether a given point sits in the solution set. This independent OpenExamPrep section uses original numbers. OpenExamPrep is not a College Board partner and does not claim official approval, review, partnership, or exact equivalence with College Board materials.
Algebra remains the largest Math domain (16–18 operational questions, about 42.5%). Inequalities show up as clean one-variable solves, as "which point works," and as in-context at least / at most stories. About 30% of Math is in context. A calculator is allowed on every Math question. Desmos can graph y ≤ mx + b and shade; still write the inequality on scratch paper first so you know whether the line is solid and which side you meant. This test does not assess trigonometry, circle theorems, or rational or radical equations, so the boundary you graph here is still a line, not a curve.
One variable: isolate, then read the symbol
The moves match 9.1 Linear Equations in One Variable—add, subtract, multiply, divide on both sides—except one extra rule:
If you multiply or divide both sides by a negative number, reverse the inequality.
Why the reverse happens: 2 < 5 is true. Multiply both sides by -1 and you get -2 and -5. On the number line, -2 is to the right of -5, so -2 > -5. The inequality symbol flipped to keep the sentence true. Adding or subtracting a negative does not reverse the symbol. Only multiply or divide by a negative.
| Symbol | Words | Boundary included? | Number-line mark | Two-variable boundary |
|---|---|---|---|---|
| < | less than | No | Open circle | Dashed line |
| > | greater than | No | Open circle | Dashed line |
| ≤ | less than or equal / at most | Yes | Closed circle | Solid line |
| ≥ | greater than or equal / at least | Yes | Closed circle | Solid line |
Example. Solve 3x - 5 > 7.
Add 5: 3x > 12. Divide by 3 (positive, do not flip): x > 4.
Check a number greater than 4, such as 5: 3(5) - 5 = 10 > 7, true. Check the boundary 4: 3(4) - 5 = 7, and 7 > 7 is false, so 4 is not included. Check 3: 9 - 5 = 4 > 7 is false.
Example. Solve 4x + 2 ≤ 18.
Subtract 2: 4x ≤ 16. Divide by 4: x ≤ 4.
Check 4: 4(4) + 2 = 16 + 2 = 18 ≤ 18, true. Check 5: 20 + 2 = 22 ≤ 18, false. Check 0: 2 ≤ 18, true. Every number 4 or smaller works.
Example (the reverse). Solve -2x + 3 ≤ 11.
Subtract 3: -2x ≤ 8. Divide by -2 and reverse: x ≥ -4.
Check the boundary x = -4: -2(-4) + 3 = 8 + 3 = 11, and 11 ≤ 11 is true. Check x = -5 (left of -4): -2(-5) + 3 = 10 + 3 = 13, and 13 ≤ 11 is false. Check x = 0: 3 ≤ 11, true. The solution runs from -4 to the right, including -4. Forgetting the reverse would produce x ≤ -4, which fails the x = 0 check immediately.
Example (negative distribute). Solve -3(x - 2) > 9.
Distribute: -3x + 6 > 9. Subtract 6: -3x > 3. Divide by -3 and reverse: x < -1.
Check x = -2: -3(-2 - 2) = -3(-4) = 12 > 9, true. Check x = -1: -3(-1 - 2) = -3(-3) = 9 > 9, false. Check x = 0: -3(0 - 2) = -3(-2) = 6 > 9, false.
If you drop the reverse after dividing by -3, you get x > -1, which makes x = 0 look like a solution even though 6 > 9 is false.
Compound inequalities at 8/9 level
A compound inequality can squeeze x between two bounds. Perform the same operation on all three parts.
Example. Solve 2 < x + 1 ≤ 6.
Subtract 1 from all three parts: 1 < x ≤ 5.
Check x = 3: 2 < 4 ≤ 6, true. Check x = 1: 2 < 2 ≤ 6, false (the left end is strict). Check x = 5: 2 < 6 ≤ 6, true (the right end includes equality). Check x = 6: 2 < 7 ≤ 6, false.
Example. Solve -3 ≤ 2x + 1 < 7.
Subtract 1: -4 ≤ 2x < 6. Divide by 2 (positive): -2 ≤ x < 3.
Check x = -2: 2(-2) + 1 = -3, and -3 ≤ -3 < 7, true. Check x = 0: 1 is between -3 and 7, true. Check x = 3: 2(3) + 1 = 7, and -3 ≤ 7 < 7 is false.
If a stem splits the compound into "x > 1 and x ≤ 5," that is the same set as 1 < x ≤ 5. An "or" compound such as x < -2 or x > 5 is two separate rays; graph both if it appears, but most PSAT 8/9 inequality items are a single bound or a simple between-statement.
Two variables: which side of a line
A linear inequality in two variables, such as y ≤ 2x + 1, describes a half-plane: all points on one side of a line, and maybe the line itself.
Steps:
- Pretend the inequality is an equation and graph the boundary y = 2x + 1.
- Use a solid line for ≤ or ≥ (points on the line count) and a dashed line for < or > (points on the line do not count).
- Test a point not on the line. (0, 0) is convenient when the line does not go through the origin.
- If the test point makes the inequality true, shade that side. If false, shade the other side.
Example. Graph the solution of y ≤ 2x + 1.
Boundary: y = 2x + 1, solid because of ≤. Test (0, 0): 0 ≤ 1, true. Shade the side that contains (0, 0), including the line. (0, 3): 3 ≤ 1, false — not in the solution. (1, 3): 3 ≤ 2(1) + 1 = 3, and 3 ≤ 3, true — on the line, allowed because of ≤.
Example. Graph the solution of y > -x + 3.
Boundary: y = -x + 3, dashed because of >. Test (0, 0): 0 > 3, false. Shade the side that does not contain (0, 0). (0, 4): 4 > 3, true. (2, 0): 0 > -2 + 3 = 1, and 0 > 1, false. (3, 0) sits on the line because 0 = -3 + 3; a dashed line means (3, 0) is not a solution.
Desmos check: type y > -x + 3 and confirm (0, 4) sits in the shaded region and (0, 0) does not. Then still test by hand once so a mistyped inequality does not fool you. Graphing the related line without shading is the skill in 9.2 Linear Equations in Two Variables.
A system of two linear inequalities is the overlap of two half-planes. PSAT 8/9 items more often give one inequality and ask whether a point works. If two inequalities appear, a point must satisfy both. Example: y ≥ x and x ≥ 0. The point (2, 3) works because 3 ≥ 2 and 2 ≥ 0. The point (2, 1) fails because 1 ≥ 2 is false. The point (-1, 0) fails x ≥ 0.
Word problems: at least and at most
| Phrase | Symbol |
|---|---|
| at least, no fewer than, minimum | ≥ |
| at most, no more than, maximum | ≤ |
| more than, greater than | > |
| fewer than, less than | < |
Example. A food truck sells tacos for $4 and drinks for $2. The crew wants at least $60 in sales. Let t be the number of tacos and d the number of drinks.
4t + 2d ≥ 60 Divide by 2: 2t + d ≥ 30.
Is (10, 8) in the solution set? 4(10) + 2(8) = 40 + 16 = 56, and 56 ≥ 60 is false. Is (12, 8)? 48 + 16 = 64 ≥ 60, true. Is (15, 0)? 60 + 0 = 60 ≥ 60, true — "at least" includes equality. Is (14, 1)? 56 + 2 = 58 ≥ 60, false.
Example. A student may work at most 12 hours this week. She already worked 5 hours. Let h be additional hours. Then 5 + h ≤ 12, so h ≤ 7. Seven more hours is allowed; eight is not: 5 + 8 = 13, which is over 12.
Example. A score must be at least 80: s ≥ 80. A score of 80 counts. A stem that says "more than 80" is s > 80, so 80 fails. Mixing those two phrases is a common trap on in-context items.
On multiple choice, plugging each ordered pair into the inequality is often faster than shading a full half-plane. On SPR you might type the boundary number (include a negative sign if needed). Do not type an inequality symbol, a percent sign, or a dollar sign in the SPR box.
What is the solution of -4x + 6 ≥ 18?
Which point is in the solution set of y > 2x - 1?
A club needs at least $90 from pins that cost $5 each and stickers that cost $2 each. Which purchase meets the goal?