solving quadratic inequalities worksheet

Guri Bee

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23-10-2024

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Mastering Quadratic Inequalities: A Step-by-Step Guide with Examples

Solving quadratic inequalities can feel daunting at first, but with the right approach, it becomes manageable. This article will guide you through the process, breaking down each step and illustrating them with practical examples.

Understanding Quadratic Inequalities

A quadratic inequality is an inequality involving a quadratic expression, often in the form of ax² + bx + c < 0 (or >, ≤, ≥). The solution set for a quadratic inequality is not a single value but a range of values that satisfy the inequality.

Let's start with a fundamental question:

Q: What is the key difference between solving a quadratic equation and a quadratic inequality?

A: While solving a quadratic equation aims to find specific values of x that make the expression equal to zero, solving a quadratic inequality determines the intervals of x where the expression is greater than or less than zero. Credit: GitHub user

Step-by-Step Guide to Solving Quadratic Inequalities

1. Rewrite the inequality in standard form: Make sure the quadratic expression is on one side of the inequality and zero is on the other side.
2. Find the roots (solutions) of the corresponding quadratic equation: Set the quadratic expression equal to zero and solve for x. This will give you the critical points that define the intervals on the number line.
3. Create a sign table (or number line): Place the roots on a number line. These roots divide the number line into intervals.
4. Test a value from each interval: Choose a test value within each interval and substitute it into the original inequality. If the inequality holds true, then all values in that interval satisfy the inequality.
5. Write the solution in interval notation: Identify the intervals where the inequality holds true and express them in interval notation.

Example: Solve the inequality x² - 3x - 4 < 0.

1. Standard Form: The inequality is already in standard form.

2. Finding roots:

• x² - 3x - 4 = 0
• (x - 4)(x + 1) = 0
• x = 4, x = -1

3. Sign Table:

4. Testing Intervals:

• x < -1: Choose x = -2. (-2)² - 3(-2) - 4 = 6 > 0 (not satisfied)
• -1 < x < 4: Choose x = 0. 0² - 3(0) - 4 = -4 < 0 (satisfied)
• x > 4: Choose x = 5. 5² - 3(5) - 4 = 6 > 0 (not satisfied)

5. Solution: The solution is -1 < x < 4, which can be written in interval notation as (-1, 4).

Additional Tips:

• Understanding the graph: The graph of a quadratic equation can help you visualize the solution. The parabola will lie below the x-axis in the intervals where the inequality is less than zero.
• Handling non-strict inequalities: For inequalities with ≤ or ≥, include the roots in the solution set.
• Factoring challenges: If the quadratic expression doesn't factor easily, use the quadratic formula to find the roots. Credit: GitHub user

Key takeaways

Solving quadratic inequalities combines algebraic manipulation with the understanding of interval analysis. This article has provided a structured approach with a real-world example. By practicing these steps and gaining confidence in the process, you'll become adept at solving quadratic inequalities and their applications in various mathematical fields.