Inequalities Mastery Quiz

01

TranslationEasy

Which inequality represents: "5 less than a number $x$ is greater than 12"?

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Memory Key: "Less than" ≠ subtract left "5 less than x" → start with x, then subtract 5 → $x - 5$. Don't flip it to $5 - x$!

A $5 - x > 12$

B $x - 5 > 12$

C $x + 5 > 12$

D $x - 5 \geq 12$

02

TranslationTricky

Translate: "3 more than twice a number $n$ is at most 15".

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Memory Key: "At most" = ≤ At most = maximum = can be equal or less → use ≤. "At least" = minimum = ≥.

A $2n + 3 \leq 15$

B $2n + 3 < 15$

C $3n + 2 \leq 15$

D $2n + 3 \geq 15$

03

TranslationHard

Which inequality matches: "The quotient of $x$ and 4, decreased by 1, is no less than 7"?

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Memory Key: "No less than" = ≥ "No less than 7" means it cannot go below 7 → use ≥. Same as "at least".

A $\dfrac{x}{4} - 1 > 7$

B $\dfrac{4}{x} - 1 \geq 7$

C $\dfrac{x}{4} - 1 \geq 7$

D $\dfrac{x - 1}{4} \geq 7$

04

TranslationTricky

Which of the following is NOT equivalent to "x is no more than 9"?

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Memory Key: "No more than" synonyms "No more than", "at most", "does not exceed", "maximum" → all mean ≤.

A $x \leq 9$

B $x$ does not exceed 9

C $x < 9$

D The maximum value of $x$ is 9

05

TranslationHard

Translate: "Half of the sum of $x$ and 6 is less than 10".

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Memory Key: "Half of the SUM" → parentheses! You add first, then divide. $\frac{1}{2}(x+6)$ ≠ $\frac{x}{2} + 6$

A $\dfrac{x}{2} + 6 < 10$

B $\dfrac{1}{2}(x + 6) < 10$

C $\dfrac{1}{2}(x + 6) \leq 10$

D $x + 6 < 5$

06

Real WorldVariable

A sign says: "Maximum capacity: 300 people."
Let $p$ = number of people in the building. Which inequality models this?

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Memory Key: "Maximum" → ≤ (can equal max) Maximum 300 means you can have exactly 300, but not 301. Equal IS allowed → ≤.

A $p < 300$

B $p \leq 300$

C $p \geq 300$

D $p = 300$

07

Real WorldTricky

A roller coaster requires riders to be at least 48 inches tall.
Let $h$ = a rider's height. Which inequality AND variable definition is correct?

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Memory Key: "At least" = minimum = ≥ At least 48" → $h \geq 48$. The height must be 48 or taller.

A Let $h$ = height in cm; $\;h \geq 48$

B Let $h$ = height in inches; $\;h \geq 48$

C Let $h$ = height in inches; $\;h > 48$

D Let $h$ = height in inches; $\;h \leq 48$

08

Real WorldHard

Maria wants to spend less than $50 total. She already spent $18. Let $m$ = remaining amount she can spend. Which inequality models this?

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Memory Key: total = already spent + remaining Set up: $18 + m < 50$, or solve directly: $m < 32$.

A $m < 50$

B $m < 32$

C $m \leq 32$

D $m + 50 < 18$

09

Real WorldHard

A speed limit sign says 65 MPH. A car is traveling at speed $s$. Which inequality correctly models a car that is breaking the speed limit?

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Memory Key: "Breaking" = going over = STRICT > Legal speed: $s \leq 65$. Breaking the limit: $s > 65$ (strictly greater).

A $s \leq 65$

B $s \geq 65$

C $s > 65$

D $s < 65$

10

Real WorldVariable

A student needs at least 90 points to get an A. She scored 87 on her first test. Let $s$ = her second test score needed (average of 2 tests). Which models the requirement?

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Memory Key: Average = sum ÷ count Average of 2 tests: $\frac{87 + s}{2} \geq 90$

A $87 + s \geq 90$

B $\dfrac{87 + s}{2} \geq 90$

C $\dfrac{87 + s}{2} > 90$

D $s \geq 90$

11

SolvingAdd/Subtract

Solve: $x + 7 < 3$

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Memory Key: Add/Subtract → sign stays same Subtract 7 from both sides. The inequality direction does NOT flip.

A $x < 10$

B $x > -4$

C $x < -4$

D $x \leq -4$

12

SolvingFlip the Sign!

Solve: $-3x \geq 12$

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🔥 Memory Key: FLIP when dividing/multiplying by NEGATIVE! Divide both sides by −3 → inequality flips from ≥ to ≤. This is the #1 most common mistake!

A $x \geq -4$

B $x \leq 4$

C $x \geq 4$

D $x \leq -4$

13

SolvingDivide

Solve: $\dfrac{x}{5} \leq -2$

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Memory Key: Multiply by POSITIVE → no flip Multiply both sides by 5 (positive). Sign stays ≤.

A $x \leq -7$

B $x \leq -10$

C $x \geq -10$

D $x \leq 10$

14

SolvingNegative Multiply

Solve: $-\dfrac{x}{4} > 3$

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🔥 Memory Key: FLIP when multiplying by −4 Multiply both sides by −4 → flip > to <. Result: $x < -12$.

A $x < -12$

B $x > -12$

C $x > 12$

D $x < 12$

15

SolvingTwo-Step

Solve: $2x + 3 > 11$

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Memory Key: Two-step → undo addition FIRST, then division Step 1: subtract 3 → $2x > 8$. Step 2: divide by 2 (positive, no flip) → $x > 4$.

A $x > 7$

B $x > 4$

C $x \geq 4$

D $x < 4$

16

GraphOpen/Closed

A number line shows a closed circle at 3, shading to the right. Which inequality does this represent?

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Memory Key: Open = strict (< or >), Closed = equal included (≤ or ≥) Closed dot = the number IS included = ≥ or ≤.

A $x > 3$

B $x \geq 3$

C $x \leq 3$

D $x < 3$

17

GraphTricky

Which value is NOT a solution of $x \leq -2$?

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Memory Key: Substitute to test! Plug each option into $x \leq -2$. If the statement is FALSE, that's your answer.

A $-5$

B $-2$

C $-10$

D $0$

18

GraphHard

After solving $5 - 2x > 9$, you graph the solution on a number line. Which description is correct?

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🔥 Remember: Solve first, FLIP when dividing by negative! $5 - 2x > 9$ → $-2x > 4$ → divide by −2, flip → $x < -2$. Open circle at −2, shade LEFT.

A Open circle at $-2$, shading to the right

B Closed circle at $-2$, shading to the left

C Open circle at $-2$, shading to the left

D Open circle at $2$, shading to the left

19

ChallengeWord Problem

A taxi charges a flat fee of $3 plus $2 per mile. Jenna has at most $15 to spend. What is the maximum number of miles she can ride? Set up the inequality.

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Memory Key: flat fee + rate × miles ≤ budget $3 + 2m \leq 15$ → $2m \leq 12$ → $m \leq 6$. She can ride at most 6 miles.

A $2m \leq 15$, so $m \leq 7.5$

B $3 + 2m \leq 15$, so $m \leq 6$

C $3 + 2m < 15$, so $m < 6$

D $3m + 2 \leq 15$, so $m \leq \frac{13}{3}$

20

ChallengeBoss Level

Which of the following inequalities has the solution $x > -3$?

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Memory Key: Work backwards — substitute $x = -3$ as boundary test Try $x = 0$ (should work) and $x = -5$ (should NOT work) in each option to verify.

A $-2x > 6$

B $-2x < 6$

C $2x < -6$

D $2x > -6$

InequalitiesMastery Quiz

Inequalities
Mastery Quiz