PSAT 8/9 Math: Linear Inequalities

Linear inequalities describe a range of possible values instead of a single answer, and the PSAT 8/9 tests this through graphs, systems, compound inequalities, and word problems. This free, complete lesson covers reading and testing inequality graphs, systems of inequalities, compound inequalities, and translating words into inequalities. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback and full explanations. Everything here is free, and you can keep practicing afterward with the linked quiz below.

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1. Reading and Testing Inequality Graphs

A graphed inequality shades a region of the coordinate plane, every point in the shaded region satisfies the inequality. To test whether a given point works, substitute its coordinates into the inequality and check if the resulting statement is true.

Worked Example: The inequality y ≤ 2x + 5 is graphed with shading below the line. Does the point (3, 4) satisfy the inequality?

Substitute: 4 ≤ 2(3) + 5 → 4 ≤ 11.

True, so (3, 4) satisfies the inequality.

1. The inequality y ≤ 2x + 5 is graphed. Does the point (3, 4) satisfy the inequality? (see worked example above)

Explanation: Substituting x = 3, y = 4: 4 ≤ 2(3) + 5 = 11. Since 4 ≤ 11 is true, the point satisfies the inequality.

2. Which point satisfies both y ≤ 2x + 5 and y ≥ −x − 2?

Explanation: Test (1, 3): 3 ≤ 2(1)+5=7 TRUE, and 3 ≥ −1−2=−3 TRUE. Both pass. The other points each fail at least one inequality.

3. The inequality y > 3x − 4 is graphed with a dashed line and shading above. Does the point (2, 2) satisfy the inequality?

Explanation: Substituting: 2 > 3(2) − 4 = 2. Since the inequality is strict (>, not ≥), and 2 is not strictly greater than 2, the point does not satisfy the inequality (it lies exactly on the boundary line).

2. Systems of Inequalities

A system of inequalities requires a point to satisfy every inequality simultaneously. To test a point against a system, check it against each inequality one at a time, it only works if it passes all of them.

Worked Example: Which point satisfies both y < 4x + 2 and x + y ≥ 3: (3, 15) or (1, 3)?

Test (3, 15): 15 < 4(3)+2=14? FALSE.

Test (1, 3): 3 < 4(1)+2=6? TRUE. And 1+3=4 ≥ 3? TRUE.

(1, 3) satisfies both inequalities.

1. Which point satisfies both y < 4x + 2 and x + y ≥ 3: (3, 15) or (1, 3)? (see worked example above)

Explanation: (3, 15) fails the first inequality (15 is not less than 14). (1, 3) passes both: 3 < 6 is true, and 1 + 3 = 4 ≥ 3 is true.

2. A system requires y ≤ x + 6 and y ≥ 2x − 1. Does the point (2, 3) satisfy this system?

Explanation: Check y ≤ x + 6: 3 ≤ 2 + 6 = 8, TRUE. Check y ≥ 2x − 1: 3 ≥ 2(2) − 1 = 3, TRUE (since ≥ includes equal). Both pass.

3. A system requires x > 0 and y > 0 and x + y < 10. Which point satisfies all three inequalities?

Explanation: (3, 4): x > 0 TRUE, y > 0 TRUE, 3+4=7 < 10 TRUE. All three pass. The other options fail at least one condition (negative x, sum ≥ 10, or x = 0).

3. Compound Inequalities

A compound inequality bounds a quantity between two values, like 0.4 ≤ p ≤ 0.7. To build one from a formula, substitute both boundary values of the input into the expression to get the corresponding bounds for the output.

Worked Example: The calories burned during a workout are modeled by T = 140p + 50, where p is intensity, with 0.4 ≤ p ≤ 0.7. Write a compound inequality for T.

At p = 0.4: T = 140(0.4) + 50 = 56 + 50 = 106.

At p = 0.7: T = 140(0.7) + 50 = 98 + 50 = 148.

106 ≤ T ≤ 148

1. The calories burned during a workout are modeled by T = 140p + 50, where p is intensity, with 0.4 ≤ p ≤ 0.7. Write a compound inequality for T. (see worked example above)

Explanation: At p = 0.4, T = 140(0.4)+50 = 106. At p = 0.7, T = 140(0.7)+50 = 148. So 106 ≤ T ≤ 148.

2. A cyclist rides between 15 and 18 miles per hour for exactly 5 hours. Write a compound inequality for the total distance d.

Explanation: Distance = rate × time. At 15 mph: 15 × 5 = 75. At 18 mph: 18 × 5 = 90. So 75 ≤ d ≤ 90.

3. A number n satisfies −3 ≤ n ≤ 5. What are the bounds for the expression 2n + 1?

Explanation: At n = −3: 2(−3)+1 = −5. At n = 5: 2(5)+1 = 11. So −5 ≤ 2n + 1 ≤ 11.

4. Translating Words into Inequalities

Certain phrases signal specific inequality symbols: "at least" means ≥, "at most" means ≤, "more than" means >, and "fewer than" or "less than" means <. Translate each part of a sentence into its matching symbol or expression before writing the full inequality.

Worked Example: A rectangle's perimeter is at most 180 inches, its length is 40 inches, and its width is w. Write an inequality for w.

Perimeter = 2(length + width) = 2(40 + w).

"At most 180" means: 2(40 + w) ≤ 180

1. A rectangle's perimeter is at most 180 inches, its length is 40 inches, and its width is w. Write an inequality for w. (see worked example above)

Explanation: "At most" means less than or equal to, and perimeter = 2(length + width), so 2(40 + w) ≤ 180.

2. The minimum value of x is 8 more than 3 times a number k. Write an inequality for x.

Explanation: "Minimum value" means x is at least that value, so x ≥ ... "8 more than 3 times a number k" translates to 3k + 8. Combined: x ≥ 3k + 8.

3. Four consecutive even integers are x, x+2, x+4, x+6. The sum of the first and third integers is at least 30. Write an inequality for x.

Explanation: The first integer is x, and the third is x + 4. Their sum is x + (x + 4) = 2x + 4. "At least 30" means ≥ 30, so 2x + 4 ≥ 30.

Common Mistakes to Avoid

Forgetting to flip the inequality sign when multiplying or dividing by a negative number, this rule applies every time, even inside a word problem.
Mixing up strict (<, >) and inclusive (≤, ≥) inequalities, a point exactly on the boundary line satisfies an inclusive inequality but not a strict one.
Checking a point against only one inequality in a system and stopping there, a point must satisfy every inequality in the system to be a valid solution.
Misreading "at least" and "at most." "At least" means a minimum (≥), while "at most" means a maximum (≤), these are easy to reverse under time pressure.
Substituting only one boundary value when building a compound inequality from a formula, both the minimum and maximum of the input must be substituted to find both bounds of the output.

Frequently Asked Questions

How is testing a point in an inequality different from testing it in an equation?

With an equation, substituting a point either makes both sides exactly equal or it doesn't. With an inequality, substituting a point produces a true or false statement (like 4 ≤ 11), and the point is a solution only when that statement is true.

Why does dividing by a negative number flip the inequality sign?

Dividing both sides of an inequality by a negative number reverses the order of the numbers on the number line, for example, 2 < 5 is true, but dividing both sides by −1 gives −2 and −5, where −2 > −5, not −2 < −5. Flipping the sign keeps the statement accurate.

Where can I practice more problems like these?

The Linear Inequalities quizzes in the PSAT 8/9 Math Question Bank include additional original problems on this topic, along with quizzes covering every other Heart of Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill tested on the PSAT 8/9. You can also browse the full PSAT 8/9 Math Question Bank.