Quadratic inequalities are an important part of algebra that extend beyond solving equations. Instead of finding just the roots of a quadratic, inequalities help you find intervals where the expression is positive or negative. Mastering these step-by-step methods can greatly improve your confidence when tackling quadratic problems in school, exams, or real-life scenarios.
In this article, we’ll walk you through a clear, step-by-step guide to solving quadratic inequalities, complete with examples and practical tips.
🔍 What Is a Quadratic Inequality?
A quadratic inequality involves a quadratic expression (like ax² + bx + c) that is compared using an inequality symbol. The standard forms include:
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ax² + bx + c > 0
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ax² + bx + c < 0
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ax² + bx + c ≥ 0
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ax² + bx + c ≤ 0
The goal is to find the values of x that make the inequality true, typically expressed as intervals on a number line.
🧩 Step-by-Step Method to Solve Quadratic Inequalities
Let’s explore the method step-by-step with examples.
✅ Step 1: Write the Inequality in Standard Form
Ensure the inequality is in the form:
ax² + bx + c [inequality] 0
Move all terms to one side of the inequality if needed so the right-hand side is zero.
✅ Step 2: Solve the Corresponding Quadratic Equation
Replace the inequality sign with an equal sign and solve the quadratic equation:
ax² + bx + c = 0
Use factoring, completing the square, or the quadratic formula to find the roots (critical points). These divide the number line into intervals.
✅ Step 3: Mark the Roots on a Number Line
Plot the roots on a number line. These points divide the line into intervals. For example, if the roots are x = r₁ and x = r₂, the intervals would be:
(-∞, r₁), (r₁, r₂), (r₂, ∞)
✅ Step 4: Choose a Test Point in Each Interval
Pick one number from each interval and substitute it into the original quadratic expression. This will tell you whether the expression is positive or negative in that interval.
✅ Step 5: Select the Intervals That Satisfy the Inequality
Use the signs from your test points to determine which intervals satisfy the inequality.
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If the inequality is > 0 or ≥ 0, choose intervals where the expression is positive
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If it is < 0 or ≤ 0, choose intervals where it is negative
Be careful with ≥ or ≤ — include the roots if appropriate.
🔢 Example 1: Solve x² – 5x + 6 > 0
Step 1: Standard Form
Already in the form:
x² – 5x + 6 > 0
Step 2: Solve the Equation
x² – 5x + 6 = 0
(x – 2)(x – 3) = 0
So, roots: x = 2 and x = 3
Step 3: Intervals
(-∞, 2), (2, 3), (3, ∞)
Step 4: Test Points
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x = 1 in (-∞, 2): (1)² – 5(1) + 6 = 2 → Positive
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x = 2.5 in (2, 3): (2.5)² – 5(2.5) + 6 = -0.25 → Negative
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x = 4 in (3, ∞): (4)² – 5(4) + 6 = 2 → Positive
Step 5: Choose Intervals
We want where expression > 0 → Positive intervals:
(-∞, 2) ∪ (3, ∞)
Do not include the roots since it’s a strict inequality (>).
🧠 Helpful Tips
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Use a sign chart or graph to visualize the inequality.
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Always check your test points — incorrect substitutions lead to wrong intervals.
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If there are no real roots, the expression doesn’t cross the x-axis. Analyze the sign of a (the coefficient of x²) to determine if the expression is always positive or negative.
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For perfect square quadratics like (x – 2)² ≤ 0, remember the expression is zero only at x = 2.
✅ Final Checklist for Solving Quadratic Inequalities
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Is the inequality written with zero on one side?
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Have you correctly solved the quadratic equation?
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Are the roots marked clearly on a number line?
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Did you test each interval carefully?
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Have you written your solution using proper interval notation?
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Did you check whether to include or exclude endpoints?
🎯 Conclusion
Solving quadratic inequalities doesn’t have to be confusing. By breaking the process into clear steps—standardizing the inequality, solving the equation, testing intervals, and interpreting the results—you can solve any quadratic inequality with confidence.
Whether you’re preparing for exams or solving real-world problems, this step-by-step method is a powerful skill to have in your algebra toolkit.
