Understanding quadratic inequalities takes practice. Solving them involves finding the range of values where a quadratic expression is greater or less than zero.
This guide includes the best practice questions on quadratic inequalities to help you build skill and confidence. Each question checks your ability to solve, analyze, and interpret the inequality correctly.
Key Steps to Remember
Before starting the questions, here’s a quick reminder of the process:
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Rearrange the inequality into standard form
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Solve the related quadratic equation
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Identify critical points
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Test intervals between the points
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Write the correct solution set
Now, let’s dive into the practice problems.
Practice Question 1
Solve:
x² – 4x – 5 > 0
Step-by-step guide:
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Factor: (x – 5)(x + 1) = 0
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Roots: x = 5, x = -1
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Intervals: x < -1, -1 < x < 5, x > 5
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Test points: x = -2, x = 0, x = 6
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True in intervals: x < -1 and x > 5
Answer:
x < -1 or x > 5
Practice Question 2
Solve:
x² + 2x – 8 ≤ 0
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Factor: (x + 4)(x – 2) = 0
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Roots: x = -4, x = 2
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Intervals: x < -4, -4 ≤ x ≤ 2, x > 2
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Test points: x = -5, x = 0, x = 3
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True in interval: -4 ≤ x ≤ 2
Answer:
-4 ≤ x ≤ 2
Practice Question 3
Solve:
x² + 6x + 10 < 0
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Try solving the equation: x² + 6x + 10 = 0
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Discriminant: b² – 4ac = 36 – 40 = -4
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No real roots
Conclusion:
The quadratic is always positive or always negative. Since it opens upward and never touches the x-axis, x² + 6x + 10 > 0 for all x.
So, it’s never less than 0.
Answer:
No solution
Practice Question 4
Solve:
x² ≥ 0
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This expression is always non-negative
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Any number squared is zero or positive
Answer:
x ∈ ℝ (All real numbers)
Practice Question 5
Solve:
x² – 9x + 20 < 0
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Factor: (x – 4)(x – 5) = 0
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Roots: x = 4, x = 5
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Intervals: x < 4, 4 < x < 5, x > 5
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Test points: x = 3, x = 4.5, x = 6
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True only in interval: 4 < x < 5
Answer:
4 < x < 5
Practice Question 6
Solve:
-x² + 2x + 3 > 0
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Rewrite: -x² + 2x + 3 = 0
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Use quadratic formula:
x = [-2 ± √(4 + 12)] / -2
x = [-2 ± √16] / -2
x = [-2 ± 4] / -2
x = 1, x = -3 -
Intervals: x < -3, -3 < x < 1, x > 1
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Test points: x = -4, x = 0, x = 2
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True in: -3 < x < 1
Answer:
-3 < x < 1
Practice Question 7
Solve:
(x + 2)(x – 7) ≤ 0
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Roots: x = -2, x = 7
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Intervals: x < -2, -2 ≤ x ≤ 7, x > 7
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Test points: x = -3, x = 0, x = 8
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True in: -2 ≤ x ≤ 7
Answer:
-2 ≤ x ≤ 7
Final Thoughts
Practicing these questions helps you master solving and interpreting quadratic inequalities. Focus on the signs, test points carefully, and understand the intervals.
These best practice questions on quadratic inequalities give you a strong foundation. Keep practicing, and solving these problems will soon feel easy and natural.
