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).

Solving a Quadratic Inequality

Example 1: x^2 + 3x - 10 < 0

1. Find Critical Values: Solve the quadratic equation by factoring:

   • Factoring gives: (x + 5)(x - 2) = 0
   • Solutions (critical values): x = -5 and x = 2

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).

3. 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.

4. Write the Solution:

   • Solution: -5 < x < 2

Example 2: x^2 - 3x - 18 > 0

1. Find Critical Values:

   • Factoring gives: (x + 3)(x - 6) = 0
   • Solutions: x = -3 and x = 6

2. Draw the Graph: Sketch the graph with crossings at -3 and 6.

   • U-shaped parabola.

3. Analyze the Graph:

   • Since it's greater than 0, look for sections above the x-axis.
   • Valid sections: x < -3 or x > 6

4. Write the Solution:

   • Solution: x < -3 OR x > 6

Example 3: x^2 - 6x + 5 ≥ 0

1. Rearranging the Inequality:

   • Move terms to one side: x^2 - 6x + 5 - 3x + 2 ≥ 0
   • Simplifies to: x^2 - 6x + 5 ≥ 0

2. Find Critical Values:

   • Factoring gives: (x - 1)(x - 5) = 0
   • Critical values: x = 1 and x = 5

3. Case Analysis Method:

   • Split x-axis into sections:
     • Section 1: x ≤ 1
     • Section 2: 1 < x < 5
     • Section 3: x ≥ 5

4. 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.

5. Write the Final Solution:

   • Solution: x ≤ 1 OR x ≥ 5

Conclusion

• 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.