3.5 Quadratic Inequalities
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Learning Objectives
- Understand what quadratic inequalities are and how they differ from linear inequalities
- Solve quadratic inequalities algebraically
- Use sign diagrams and graphs to determine solution sets
- Represent the solution set on a number line
- Apply quadratic inequalities to real-world contexts
Key Concepts
Introduction to Quadratic Inequalities
Quadratic inequalities involve expressions with a squared variable (e.g., x²) and use inequality symbols like <, >, ≤, or ≥.
They typically take forms such as x² - 4x + 3 < 0 or 2x² + 5x ≥ 3.
Unlike linear inequalities, the solution to a quadratic inequality may be a single interval, multiple intervals, or even no solution, depending on the shape and position of the parabola.
Solving Quadratic Inequalities Algebraically
To solve a quadratic inequality, follow these steps:
1. Rearrange the inequality so one side is 0: f(x) < 0, f(x) > 0, etc.
2. Solve the related quadratic equation (f(x) = 0) to find critical points (roots).
3. Use a sign diagram or graph to determine where the expression is positive or negative.
4. Write the solution set based on the inequality.
Example 1
Solve the inequality: x² - 5x + 6 < 0
Solution:
Step 1: Solve the related equation x² - 5x + 6 = 0 => (x - 2)(x - 3) = 0 => x = 2, x = 3 Step 2: Use a sign diagram or sketch the parabola opening upwards, crossing x-axis at 2 and 3. Step 3: The expression x² - 5x + 6 is less than 0 between the roots. Final solution: 2 < x < 3
Example 2
Solve the inequality: x² + x - 6 ≥ 0
Solution:
Step 1: Solve the related equation x² + x - 6 = 0 => (x - 2)(x + 3) = 0 => x = -3, x = 2 Step 2: The parabola opens upwards and crosses at -3 and 2. Step 3: The expression is ≥ 0 outside the roots. Final solution: x ≤ -3 or x ≥ 2
Representing Solutions on a Number Line
Solutions to quadratic inequalities can be represented as intervals on a number line.
Use open circles for < or >, and closed circles for ≤ or ≥.
Arrows or shaded regions indicate intervals where the inequality holds true.
Example 1
Represent the solution: x² - 5x + 6 < 0
Solution:
The solution 2 < x < 3 is represented by: - Open circles at 2 and 3 - A shaded region between 2 and 3
Example 2
Represent the solution: x² + x - 6 ≥ 0
Solution:
The solution x ≤ -3 or x ≥ 2 is represented by: - Closed circles at -3 and 2 - Shaded regions extending left from -3 and right from 2
Quadratic Inequalities with No Real Solutions
If the related quadratic equation has no real roots (i.e., the discriminant is negative), the parabola does not cross the x-axis.
The inequality is either always true or always false, depending on the sign of the quadratic expression.
Example 1
Solve the inequality: x² + 4x + 5 > 0
Solution:
Step 1: Discriminant D = 4² - 4×1×5 = 16 - 20 = -4 (no real roots). Step 2: Since the parabola opens upwards (positive x²), it's always above the x-axis. Final solution: x ∈ ℝ (all real numbers)
Example 2
Solve the inequality: x² + 4x + 5 < 0
Solution:
Step 1: Discriminant D = -4 → no real roots Step 2: Parabola opens upwards, never touches or crosses x-axis Final solution: No solution (∅)
Applications of Quadratic Inequalities
Quadratic inequalities are used in real-world contexts such as physics (projectile motion), business (profit maximization), and engineering (safety margins).
They help identify intervals where a certain condition is satisfied, such as when an object is above a certain height or when cost remains under a limit.
Example 1
A ball is thrown upward with height h(t) = -5t² + 20t. When is the ball at or above 15 meters?
Solution:
Step 1: Set up inequality: -5t² + 20t ≥ 15 Step 2: Rearrange: -5t² + 20t - 15 ≥ 0 Divide by -5 (reverse inequality): t² - 4t + 3 ≤ 0 Solve: (t - 1)(t - 3) ≤ 0 Solution: 1 ≤ t ≤ 3 The ball is at or above 15 meters between 1 and 3 seconds after it is thrown.
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