Quadratic inequalities are a vital part of algebra that help us understand ranges of values where a quadratic expression holds true. Unlike quadratic equations that provide specific roots, quadratic inequalities describe intervals where the quadratic expression is positive, negative, or zero.
Graphing these inequalities on a number line is an excellent way to visualize their solution sets and make sense of the results. In this blog post, we’ll explain how to graph quadratic inequalities on the number line, step by step, with clear examples.
🔍 What Is a Quadratic Inequality?
A quadratic inequality involves a quadratic expression related by an inequality sign:
ax² + bx + c > 0
ax² + bx + c < 0
ax² + bx + c ≥ 0
ax² + bx + c ≤ 0
Our goal is to find all values of x that satisfy the inequality and represent them graphically.
🎯 Why Graph on a Number Line?
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It provides a visual representation of solution intervals.
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Makes it easier to understand where the quadratic is positive or negative.
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Helps distinguish between strict (>) and inclusive (≥) inequalities.
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Useful for communicating solutions clearly and intuitively.
🧩 Step-by-Step Guide to Graph Quadratic Inequalities
Step 1: Rewrite the Inequality in Standard Form
Ensure the inequality is in the form:
ax² + bx + c [inequality] 0
Step 2: Find the Roots of the Corresponding Quadratic Equation
Solve the equation:
ax² + bx + c = 0
Find roots r₁ and r₂ (real or complex).
Step 3: Mark the Roots on the Number Line
Plot the roots in increasing order. These points divide the number line into intervals.
Step 4: Determine the Sign of the Quadratic Expression in Each Interval
Choose a test point in each interval and substitute into ax² + bx + c.
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Mark the interval as “+” if positive.
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Mark the interval as “−” if negative.
Step 5: Decide Which Intervals to Include Based on the Inequality
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For > or <, select intervals where the expression is strictly positive or negative.
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For ≥ or ≤, include roots with solid dots to show they are part of the solution.
Step 6: Graph the Solution Set on the Number Line
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Shade the intervals that satisfy the inequality.
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Use open circles at roots if the inequality is strict.
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Use closed circles if roots are included.
🔢 Example: Graph the Inequality
x² – 5x + 6 ≤ 0
Step 1: Standard form
Already in standard form.
Step 2: Find roots
Factor:
(x – 2)(x – 3) = 0
Roots: 2 and 3
Step 3: Mark roots
Place points at 2 and 3 on the number line.
Step 4: Test intervals
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For x < 2, test x=1:
1² – 5(1) + 6 = 1 – 5 + 6 = 2 > 0 → positive -
For 2 < x < 3, test x=2.5:
2.5² – 5(2.5) + 6 = 6.25 – 12.5 + 6 = -0.25 < 0 → negative -
For x > 3, test x=4:
4² – 5(4) + 6 = 16 – 20 + 6 = 2 > 0 → positive
Step 5 & 6: Select intervals and graph
Inequality is ≤ 0, so choose the interval where expression ≤ 0:
[2, 3]
Graph this by shading the section between 2 and 3 and placing closed dots at 2 and 3.
📝 Additional Tips
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For no real roots, analyze the sign of a to decide if the quadratic is always positive or negative.
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When roots are complex, the quadratic does not cross the x-axis, so the entire number line might be the solution or no solution depending on a and the inequality.
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Always double-check test points and be mindful of strict versus inclusive inequalities.
🎯 Conclusion
Graphing quadratic inequalities on the number line provides a clear and intuitive way to represent solutions visually. This approach helps students and professionals alike to better understand the intervals where the quadratic expression meets the inequality conditions.
