Lesson 1.21

Quadratic Inequalities

We know when the projectile hits the ground (). But when is it above 100 feet? That requires solving an inequality like .

Introduction

Solving is trickier than linear inequalities. You can't just isolate . You have to find the "critical points" and test the regions between them.

Past Knowledge

You can find the roots (where it equals zero).

Today's Goal

Use a number line to determine where the parabola is positive (above axis) or negative (below axis).

Future Success

This "Test Point Method" is the exact same logic used in Calculus for finding where functions increase or decrease.

Key Concepts

1. The Test Point Method

  1. Find Critical Points: Solve . Mark them on a number line.
  2. Test Intervals: Pick a number in each region (like 0, 10, -10). Plug it in.
  3. Sign Check: Is the result Positive (+) or Negative (-)? Does that match your inequality ( or )?
  4. Shade: Mark the regions that work.

2. Visual Rule of Thumb

For (parabola opens up):

Less Than ()

Usually "Between" the roots. (The dip below water).

Greater Than ()

Usually "Outside" the roots. (The arms going up).

Worked Examples

Example 1: Less Than (Between)

Basic

Solve .

1

Find Roots

2

Test Regions

  • Try : (Positive) - No.
  • Try : (Negative) - YES.
  • Try : (Positive) - No.

Solution:

Example 2: Greater Than (Outside)

Intermediate

Solve .

1

Find Roots

Factor:

Roots at and .
2

Test Regions

  • : (YES)
  • : (No)
  • : (YES)

Solution:

Note square brackets because includes the roots.

Example 3: All or Nothing

Advanced

Solve .

1

Analyze

is always positive or zero. Adding 25 makes it definitely positive.

2

Conclusion

It can never be less than zero.

Result: No Solution.

Common Pitfalls

The Test Point

Do NOT plug the test point into the equality (). Plug it into the inequality to see if it is True or False.

Brackets vs. Parentheses

use parentheses . use brackets . Infinity ALWAYS uses parentheses.

Real-Life Applications

"For how many seconds is the rocket above 1000 feet?" This is an inequality problem. You find the two times it hits 1000ft (going up and coming down), and the time between those points is your answer.

Practice Quiz

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