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Quizzes > High School Quizzes > Mathematics

Inequality Word Problems Practice Test

Practice solving tricky inequalities with answer walkthrough

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art depicting a trivia quiz on mastering inequality problems for high school students.

Solve for x: 3x + 5 < 20
x < 15
x < 5
x > 15
x > 5
Subtracting 5 from both sides gives 3x < 15. Dividing by 3 results in x < 5, which is the correct solution.
Solve for x: 2x - 4 ≤ 10
x < 7
x ≤ 7
x > 7
x ≥ 7
Adding 4 to both sides results in 2x ≤ 14. Dividing by 2 gives x ≤ 7, which is the correct answer.
Solve for x: -2x > 8
x ≥ -4
x > -4
x ≤ -4
x < -4
Dividing both sides by -2 requires reversing the inequality, leading to x < -4. This is why x < -4 is correct.
Which inequality represents: 'A number decreased by 4 is less than 10'?
4 - x < 10
10 - x < 4
x + 4 < 10
x - 4 < 10
The phrase 'a number decreased by 4' is written as x - 4. Since it is less than 10, the correct inequality is x - 4 < 10.
Find the solution set for: x + 3 > 7
x ≥ 4
x > 4
x ≤ 4
x < 4
Subtracting 3 from both sides results in x > 4. This clearly shows the correct solution to the inequality.
Solve the inequality: 5x - 2 < 3x + 6
x > 4
x < 4
x ≥ 4
x ≤ 4
Subtracting 3x from both sides gives 2x - 2 < 6, and adding 2 results in 2x < 8. Dividing by 2 yields x < 4.
Solve for x: 4(x + 2) ≥ 20
x ≤ 3
x < 3
x ≥ 3
x > 3
Expanding the expression results in 4x + 8 ≥ 20. Subtracting 8 gives 4x ≥ 12, and dividing by 4 gives x ≥ 3.
If 3x + 4 < 19, what is the greatest whole number x can be?
6
5
4
3
Subtracting 4 from both sides gives 3x < 15, so dividing by 3 results in x < 5. The greatest whole number less than 5 is 4.
Solve: -3x + 7 ≤ 1
x ≥ 2
x < 2
x > 2
x ≤ 2
Subtracting 7 from both sides results in -3x ≤ -6. Dividing by -3 (and flipping the inequality) gives x ≥ 2.
Solve for x: 2(x - 3) > 8
x ≤ 7
x > 7
x ≥ 7
x < 7
Expanding the left side gives 2x - 6 > 8. Adding 6 to both sides results in 2x > 14, and dividing by 2 yields x > 7.
A store sells notebooks for $2 each and pens for $1 each. If your total cost must be more than $10, which inequality represents purchasing n notebooks and p pens?
p - 2n > 10
2n + p < 10
2n - p > 10
2n + p > 10
The total cost is calculated by multiplying the number of notebooks by $2 and pens by $1. The inequality 2n + p > 10 correctly represents the condition that the total cost must exceed $10.
The cost for a taxi ride is a flat fee of $3 plus $2 per mile. If a rider has at most $15 to spend, which inequality represents the maximum miles m they can ride?
3 + 2m ≤ 15
3 - 2m ≤ 15
3 + 2m < 15
2m - 3 ≤ 15
The total cost of the taxi ride is given by 3 + 2m. Since the rider can spend at most $15, the inequality 3 + 2m ≤ 15 is correct.
A school club raised funds with a bake sale. The number of cookies sold, c, must satisfy 5c - 10 ≥ 40. What is the minimum number of cookies sold?
10
11
9
8
Adding 10 to both sides results in 5c ≥ 50, and dividing by 5 gives c ≥ 10. This means at least 10 cookies were sold.
Solve for x: 7 - x < 2
x < 5
x > 5
x ≥ 5
x ≤ 5
Subtracting 7 from both sides gives -x < -5. Multiplying by -1 (and reversing the inequality) results in x > 5.
A gym membership costs a monthly fee of $25 plus an activation fee of $50. If you have no more than $150 to spend, which inequality represents the number of months m you can pay for?
25m + 50 ≤ 150
25m + 50 < 150
25m + 50 ≥ 150
25m - 50 ≤ 150
The total cost is 25m + 50 and it must be at most $150. The inequality 25m + 50 ≤ 150 correctly represents this scenario.
Solve the inequality: 4 - 3(2 - x) ≥ 7
x ≤ 3
x < 3
x ≥ 3
x > 3
First, distribute -3 over (2 - x) to get 4 - 6 + 3x, which simplifies to 3x - 2. The inequality then becomes 3x - 2 ≥ 7, so adding 2 gives 3x ≥ 9 and dividing by 3 results in x ≥ 3.
Solve for x: -4(2x - 5) < 8
x < 1.5
x ≤ 1.5
x ≥ 1.5
x > 1.5
Expanding the left side gives -8x + 20 < 8. Subtracting 20 from both sides results in -8x < -12. Dividing by -8 (and reversing the inequality) yields x > 1.5.
A shop offers a discount such that the total price P for buying x items at $10 each is given by P = 10x - 3, and the sale price must be at least $50. What is the minimum number of items you must buy?
x > 5
x > 6
x ≥ 6
x ≥ 5
Setting up the inequality, 10x - 3 ≥ 50, we add 3 to obtain 10x ≥ 53. Dividing by 10 gives x ≥ 5.3, meaning the smallest whole number of items is 6.
Solve the inequality: |2x - 3| < 5
-1 < x < 4
x < 4
x > -1
-1 ≤ x ≤ 4
The inequality |2x - 3| < 5 can be rewritten as -5 < 2x - 3 < 5. Adding 3 to each part gives -2 < 2x < 8, and dividing by 2 results in -1 < x < 4.
A membership plan requires that the number of months m you subscribe satisfies 3m + 2 > 20. What is the smallest whole number of months required?
8
7
6
5
Subtracting 2 from both sides gives 3m > 18, so dividing by 3 results in m > 6. Since m must be a whole number, the smallest valid value is 7.
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Study Outcomes

  1. Analyze real-world scenarios to formulate algebraic inequalities.
  2. Apply algebraic techniques to isolate and solve inequalities.
  3. Evaluate the validity of solutions within given constraints.
  4. Interpret the implications of inequality solutions in practical contexts.
  5. Synthesize problem-solving strategies to approach diverse inequality challenges.

Inequality Word Problems Worksheet Answers Cheat Sheet

  1. Master inequality symbols - Dive into >, <, ≥, and ≤ to see how they shape relationships between numbers. By memorizing these symbols inside out, you'll decode problems in a flash and boost your confidence. Writing Inequalities Cheat Sheet
  2. Translate word problems - Spot words like "at least", "more than", "no more than" and "less than" to craft the right inequality. It's like turning real-life situations into secret codes ready to solve! Inequality Word Problems
  3. Solve one-step inequalities - Use inverse operations (add, subtract, multiply, divide) to isolate your variable. It's the foundation of inequality work: master this, and bigger challenges will feel like a walk in the park. One-Step Inequality Worksheet
  4. Tackle two-step inequalities - Combine operations and remember to flip the sign when multiplying or dividing by a negative number. A little care here prevents sign flops and keeps your solutions on point. Two-Step Inequality Word Problems
  5. Graph solutions on a number line - Use open circles for strict inequalities (>, <) and closed circles for inclusive ones (≥, ≤). Seeing your solution set visually helps cement the answer in your mind. Graphing Inequalities Guide
  6. Explore compound inequalities - Combine two inequalities with "and" or "or" to express ranges or separate solution sets. These multi-part statements add depth - and a bit of fun - to your problem-solving toolkit! Compound Inequalities Tutorial
  7. Apply inequalities in real life - Budgeting, crafting recipes, and planning trips all use inequalities ("spend at least $X" or "stay under $Y"). Practicing with everyday examples brings math off the page. Real-World Inequality Applications
  8. Check your solutions - Substitute your answers back into the original inequality to confirm they work. This quick verification step catches sneaky mistakes and earns you extra study bragging rights! Solution Checking Techniques
  9. Practice multi-step word problems - Break complex scenarios into bite‑sized steps: translate, solve, then graph. With each problem, your confidence grows and the next challenge feels more doable. Multi-Step Inequality Challenges
  10. Embrace ranges of solutions - Remember, inequalities often have infinite answers represented by value ranges. Celebrate this freedom - it shows you're thinking flexibly and mastering the concept. Exploring Inequality Solutions
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