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Quizzes > High School Quizzes > Mathematics

Inequalities Practice Quiz for Exam Success

Sharpen your skills with challenging practice tests

Difficulty: Moderate
Grade: Grade 8
Study OutcomesCheat Sheet
Paper art promoting the Inequality Challenge, a practice quiz for mastering algebraic inequalities.

Solve the inequality 2x + 3 < 7. What is the solution for x?
x < 2
x > 2
x < 4
x > 4
Subtracting 3 from both sides gives 2x < 4, and dividing by 2 yields x < 2. This shows that all numbers less than 2 satisfy the inequality.
Solve the inequality x - 5 ≥ 3.
x ≥ 8
x ≤ 8
x > 3
x < 3
Adding 5 to both sides results in x ≥ 8. This is the only solution that meets the condition given by the inequality.
Which inequality symbol correctly represents 'greater than or equal to'?
≥
>
<
≤
The symbol ≥ is used to indicate 'greater than or equal to'. The other symbols either indicate a strict inequality or the reverse relation.
For the inequality -3x > 9, what is the solution for x?
x < -3
x > -3
x ≤ -3
x ≥ -3
Dividing both sides by -3 reverses the inequality sign, yielding x < -3. This reversal is a key property when dealing with negative coefficients in inequalities.
What happens to an inequality if both sides are multiplied by a negative number?
The direction of the inequality is reversed
The inequality remains unchanged
The inequality becomes an equality
The inequality becomes invalid
Multiplying or dividing both sides of an inequality by a negative number forces the inequality sign to flip. This is a fundamental rule in solving inequalities.
Solve the inequality: 4 - 2x ≤ 10.
x ≥ -3
x ≤ -3
x > -5
x < -5
Subtracting 4 from both sides gives -2x ≤ 6. Dividing by -2 reverses the inequality, resulting in x ≥ -3. This careful handling of a negative divisor is crucial.
Solve the compound inequality: -2 ≤ 3x + 1 < 8.
−1 ≤ x < 7/3
x < -1 or x ≥ 7/3
−1 < x ≤ 7/3
x ≤ -1 or x > 7/3
Subtracting 1 from all parts of the inequality yields -3 ≤ 3x < 7, and then dividing by 3 results in −1 ≤ x < 7/3. The compound inequality has both an inclusive and an exclusive boundary.
Which inequality is equivalent to 5x - 4 > 11?
x > 3
x ≥ 3
x < 3
x ≤ 3
Adding 4 to both sides gives 5x > 15, and dividing by 5 leads to x > 3. This is the precise transformation of the original inequality.
What is the solution of 2(x - 1) > 6?
x > 4
x ≥ 4
x < 4
x ≤ 4
Expanding the left side yields 2x - 2 > 6; adding 2 results in 2x > 8, and dividing by 2 gives x > 4. The strict inequality indicates that x must be greater than 4.
Solve the inequality: -4(x + 2) ≤ 8.
x ≥ -4
x ≤ -4
x > -4
x < -4
After expanding to get -4x - 8 ≤ 8, adding 8 produces -4x ≤ 16. Dividing by -4 and flipping the inequality results in x ≥ -4.
Solve the inequality: 3x + 2 < 2x + 5.
x < 3
x > 3
x ≤ 3
x ≥ 3
Subtracting 2x from both sides leads to x + 2 < 5, and further subtracting 2 gives x < 3. This is a straightforward linear inequality.
Solve the inequality: -2(-x + 3) > 8.
x > 7
x < 7
x ≥ 7
x ≤ 7
Distributing -2 gives 2x - 6 > 8; adding 6 results in 2x > 14, and dividing by 2 concludes x > 7. The steps highlight the importance of carefully handling negative signs.
Solve the inequality: ½x - 4 < 3.
x < 14
x > 14
x ≤ 14
x ≥ 14
Adding 4 to both sides yields ½x < 7, and multiplying by 2 leads to x < 14. The multiplication by a positive number keeps the inequality direction unchanged.
Solve the inequality: 3(x - 2) + 4 ≤ 2x + 1.
x ≤ 3
x > 3
x ≥ 3
x < 3
Expanding gives 3x - 6 + 4 ≤ 2x + 1, which simplifies to x - 2 ≤ 1. Adding 2 to both sides results in x ≤ 3, the correct solution.
Determine the solution for the inequality: 2 - 3x > 5.
x < -1
x > -1
x ≤ -1
x ≥ -1
Subtracting 2 from both sides yields -3x > 3, and dividing by -3 (which reverses the inequality) gives x < -1. This correctly identifies all numbers less than -1 as solutions.
Solve the absolute value inequality: |2x - 3| < 5.
-1 < x < 4
-4 < x < 1
x > -1 and x ≥ 4
x ≤ -1 or x ≥ 4
The inequality |2x - 3| < 5 is equivalent to -5 < 2x - 3 < 5. By adding 3 and then dividing by 2, the solution is found to be -1 < x < 4.
Solve the inequality: (x - 2)/(x + 3) ≥ 0, considering that x ≠ -3.
(-∞, -3) ∪ [2, ∞)
(-∞, -3] ∪ (2, ∞)
(-3, 2)
[−3, ∞)
Analyzing the critical points x = -3 (undefined) and x = 2 (zero) shows that the rational expression is nonnegative for x in (-∞, -3) and for x ≥ 2. This solution excludes x = -3 due to division by zero.
Solve the inequality: 3 - 2|x + 1| > -1.
x ∈ (-3, 1)
x ∈ [-3, 1]
x ∈ (-1, 3)
x ∈ (-∞, -3) ∪ (1, ∞)
Subtracting 3 from both sides and dividing by -2 (which reverses the inequality) leads to |x + 1| < 2. This results in the compound inequality -3 < x < 1.
Solve the inequality: 1/(x - 2) ≤ 3.
(-∞, 2) ∪ [7/3, ∞)
(-∞, 2] ∪ (7/3, ∞)
(2, 7/3)
[2, 7/3]
After rewriting the inequality as (7 - 3x)/(x - 2) ≤ 0 and finding the critical values x = 7/3 and x = 2 (where x = 2 is excluded), interval testing shows the solution to be (-∞, 2) ∪ [7/3, ∞).
If 4 < 7, what is the result of multiplying both sides by -2, considering the rules of inequalities?
-8 < -14
-8 > -14
-8 = -14
-8 ≤ -14
Multiplying both sides of the inequality 4 < 7 by -2 reverses the inequality sign, resulting in -8 > -14. This demonstrates the rule that multiplication by a negative number flips the direction of an inequality.
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Study Outcomes

  1. Understand the fundamental properties and rules of algebraic inequalities.
  2. Solve linear inequalities accurately using appropriate methods.
  3. Analyze solution sets and determine the validity of inequality solutions.
  4. Apply problem-solving strategies to assess and improve algebraic skills.
  5. Evaluate personal performance to identify areas for further review and practice.

Inequalities Quiz & Test Review Cheat Sheet

  1. Understand basic inequality symbols - Inequalities let you compare values, like who has more candies or who's older. Symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) form the backbone of these comparisons. Mastering them is like learning a secret code for math adventures! SparkNotes: Inequality Basics
  2. Learn the properties of inequalities - You can add or subtract the same number on both sides without flipping the inequality's direction, which is as fair as sharing slices of pizza evenly. Multiplying or dividing by a positive number keeps the arrow pointing the same way, but hit it with a negative and you'll send it spinning around! These rules help you manipulate and simplify inequalities confidently. Mathematical Scientist: Inequality Properties
  3. Practice solving linear inequalities - Treat a linear inequality like a mini-puzzle: isolate the variable step by step until you uncover its secret range. For instance, 3x + 5 > 11 becomes 3x > 6, so x > 2. Regular practice makes these transformations feel smoother than your favorite dance move! GeeksforGeeks: Linear Inequality Practice
  4. Understand compound inequalities - Compound inequalities bundle two comparisons into one statement using "and" or "or," like setting upper and lower bounds for your video game score. Solving - 2 < 2x + 3 ≤ 7 splits into two separate inequalities you handle individually. By weaving the solutions back together, you'll see exactly where x can roam free. GeeksforGeeks: Compound Inequalities
  5. Solve absolute value inequalities - Absolute values measure distance from zero, so |x - 3| ≥ 4 means x sits at least 4 units away from 3. You break it into x - 3 ≥ 4 or x - 3 ≤ - 4, solving each with normal inequality rules. This two‑case approach ensures you capture both ends of the number line. GeeksforGeeks: Absolute Value Strategies
  6. Graph inequalities on a number line - Visual learners rejoice: open circles show "not included" endpoints, closed circles show "included," and arrows shade the solution region. Sketching x > 2 with an open circle at 2 and shading to the right makes the answer crystal clear. This graphic method cements your algebraic work in a picture! SparkNotes: Number Line Graphing
  7. Solve quadratic inequalities - Factor expressions like x² - 4x < 3 into (x - 3)(x + 1) < 0, then test intervals around your roots to see where the product flips sign. It's like playing tug‑of‑war with the zero points - figure out which side wins! Mapping these intervals reveals exactly where x can sit. GeeksforGeeks: Quadratic Inequalities
  8. Handle rational inequalities - For expressions like (x - 2)/(x + 1) > 0, find zeros and undefined points to split the number line into test zones. Plugging sample values into each zone tells you which intervals satisfy the inequality. It's a systematic hunt that guarantees you won't miss any sneaky solution bits! GeeksforGeeks: Rational Inequality Guide
  9. Tackle polynomial inequalities - Factor high‑degree polynomials like x³ - 2x² - 3x > 0 and determine sign changes across intervals. By checking each segment, you'll pinpoint exactly where the curve stays above zero. This method turns intimidating expressions into manageable chunks! GeeksforGeeks: Polynomial Inequalities
  10. Apply inequalities in real-world problems - From defining feasible regions in linear programming to setting up budget constraints, inequalities model countless scenarios. Translating word problems into algebraic inequalities builds your problem‑solving superpowers. Master this, and you'll see math's magic lighting up real life! SparkNotes: Real‑World Inequalities
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