Quadratic inequalities are similar to quadratic equations, but instead of an equal sign, they use inequality symbols like >, <, ≥, or ≤. Solving them means finding all the values of the variable that make the inequality true.
This guide will show you exactly how to solve quadratic inequalities with clear steps, explanations, and examples. By the end, you will be confident in tackling any quadratic inequality problem.
What Is a Quadratic Inequality?
A quadratic inequality involves a quadratic expression, usually in the form:
ax² + bx + c > 0
ax² + bx + c < 0
ax² + bx + c ≥ 0
ax² + bx + c ≤ 0
Here, a, b, and c are constants, and x is the variable you want to solve for. Your goal is to find all values of x that satisfy the inequality.
Unlike equations that give specific solutions, inequalities give ranges or intervals of solutions.
Step 1: Write the Inequality in Standard Form
Make sure your inequality has zero on one side. For example:
ax² + bx + c > 0
If the inequality is not in this form, rearrange it by subtracting terms from both sides. Always work with the inequality compared to zero.
Step 2: Solve the Related Quadratic Equation
Replace the inequality sign with an equal sign to find the critical points:
ax² + bx + c = 0
Solve this quadratic equation using any method you prefer:
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Factoring
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Completing the square
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Quadratic formula
The solutions you get here are called the roots or zeros of the quadratic. These roots divide the number line into intervals.
Step 3: Identify the Intervals
Once you find the roots, mark them on a number line. These roots split the line into sections or intervals.
For example, if the roots are x = 2 and x = 5, the intervals are:
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x < 2
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2 < x < 5
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x > 5
Each interval will either satisfy the inequality or not.
Step 4: Test Each Interval
Pick any number from each interval and substitute it back into the original inequality (not the equation).
Check if the inequality is true or false for that number.
If it’s true, that entire interval is part of your solution. If false, exclude that interval.
For example, using the roots 2 and 5:
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Test x = 1 (less than 2)
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Test x = 3 (between 2 and 5)
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Test x = 6 (greater than 5)
Step 5: Write the Solution Set
Based on your test results, write the solution using inequality notation or interval notation.
If the inequality is strict (>, <), use parentheses or open intervals.
If the inequality includes equal to (≥, ≤), include the roots by using brackets or equal signs.
For example:
x < 2 or x > 5
or in interval notation:
(-∞, 2) ∪ (5, ∞)
Step 6: Graph the Solution (Optional but Helpful)
Graphing can make understanding quadratic inequalities easier.
First, graph the related quadratic function:
y = ax² + bx + c
Plot the roots on the x-axis. Then, shade the regions where the inequality is true:
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If the inequality is > or ≥, shade the area where the graph is above the x-axis.
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If the inequality is < or ≤, shade where the graph is below the x-axis.
Use a solid line on the parabola if your inequality includes equal to (≥ or ≤). Use a dashed line if it does not (>, <).
Example Problem
Let’s solve:
x² – 3x – 4 > 0
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Write the equation:
x² – 3x – 4 = 0 -
Factor the quadratic:
(x – 4)(x + 1) = 0 -
Find the roots:
x = 4 and x = -1 -
Intervals on the number line:
x < -1, -1 < x < 4, x > 4 -
Test each interval:
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For x = -2: (-2)² – 3(-2) – 4 = 4 + 6 – 4 = 6 (true)
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For x = 0: 0 – 0 – 4 = -4 (false)
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For x = 5: 25 – 15 – 4 = 6 (true)
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Write the solution:
x < -1 or x > 4
Tips for Success
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Always rewrite your inequality with zero on one side before solving.
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Carefully solve the related quadratic equation to find accurate roots.
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Test points correctly by substituting into the original inequality.
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Don’t forget to include or exclude roots depending on the inequality sign.
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Use graphing tools or draw the parabola to visualize your answers.
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Practice multiple problems to build confidence.
Final Thoughts
Now you know how to solve quadratic inequalities step by step. By finding roots, testing intervals, and understanding the solution graphically, you can solve these problems with ease.
Keep practicing and use graphs to deepen your understanding. Soon, quadratic inequalities will become second nature.
