Solving Quadratic Inequalities

Introduction to Quadratic Inequalities

A quadratic inequality has a less than (<), greater than (>), less than or equal to (≤), or greater than or equal to (≥) sign.
Example: x^2 + 3x - 10 < 0 (inequality) vs x^2 + 3x - 10 = 0 (equation).

Find Critical Values: Solve the quadratic equation by factoring:
- Factoring gives: (x + 5)(x - 2) = 0
- Solutions (critical values): x = -5 and x = 2
Draw the Graph: Sketch the graph of y = x^2 + 3x - 10.
- Crosses x-axis at critical values -5 and 2.
- U-shaped parabola (quadratic).
Analyze the Graph:
- Determine where the graph is below the x-axis (less than 0).
- The green section of the graph is between -5 and 2.
Write the Solution:
- Solution: -5 < x < 2

Find Critical Values:
- Factoring gives: (x + 3)(x - 6) = 0
- Solutions: x = -3 and x = 6
Draw the Graph: Sketch the graph with crossings at -3 and 6.
- U-shaped parabola.
Analyze the Graph:
- Since it's greater than 0, look for sections above the x-axis.
- Valid sections: x < -3 or x > 6
Write the Solution:
- Solution: x < -3 OR x > 6

Rearranging the Inequality:
- Move terms to one side: x^2 - 6x + 5 - 3x + 2 ≥ 0
- Simplifies to: x^2 - 6x + 5 ≥ 0
Find Critical Values:
- Factoring gives: (x - 1)(x - 5) = 0
- Critical values: x = 1 and x = 5
Case Analysis Method:
- Split x-axis into sections:
  - Section 1: x ≤ 1
  - Section 2: 1 < x < 5
  - Section 3: x ≥ 5
Test Each Section:
- Test a number in each section to check the validity:
  - For x ≤ 1: Choosing 0, gives valid solution.
  - For 1 < x < 5: Choosing 2, does not satisfy.
  - For x ≥ 5: Choosing 6, gives valid solution.
Write the Final Solution:
- Solution: x ≤ 1 OR x ≥ 5

Solving quadratic inequalities involves determining critical values and analyzing the graph or using case analysis.
Solutions can be expressed using AND or OR depending on the situation.
Practice with various inequalities to become proficient.