2.2.4 Quadratic Inequalities

Quadratic Inequalities

The inequality sign is unchanged by...
- Adding/subtracting a term to both sides
- Multiplying/dividing both sides by a positive term
The inequality sign flips (< changes to >) when...
- Multiplying/dividing both sides by a negative term

STEP 1: Rearrange the inequality into quadratic form with a positive squared term
- ax² + bx + c > 0
- ax² + bx + c ≥ 0
- ax² + bx + c < 0
- ax² + bx + c ≤ 0
STEP 2: Find the roots of the quadratic equation
- Solve ax² + bx + c = 0 to get x₁ and x₂ where x₁ < x₂
STEP 3: Sketch a graph of the quadratic and label the roots
- As the squared term is positive it will be concave up so "U" shaped
STEP 4: Identify the region that satisfies the inequality
- If you want the graph to be above the x-axis then choose the region to be the two intervals outside of the two roots
- If you want the graph to be below the x-axis then choose the region to be the interval between the two roots
- For ax² + bx + c > 0
  - The solution is x < x₁ or x > x₂
- For ax² + bx + c ≥ 0
  - The solution is x ≤ x₁ or x ≥ x₂
- For ax² + bx + c < 0
  - The solution is x₁< x < x₂
- For ax² + bx + c ≤ 0
  - The solution is x₁≤ x ≤ x₂

The safest way is by following the steps above
- Expand and rearrange
A common mistake is writing $x - h < \pm \sqrt{n}$ or $x - h > \pm \sqrt{n}$
- This is NOT correct!
The correct solution to (x - h)² < n is
- $|x - h| < \sqrt{n}$ which can be written as $- \sqrt{n} < x - h < \sqrt{n}$
- The final solution is $h - \sqrt{n} < x < h + \sqrt{n}$
The correct solution to (x - h)² > n is
- $|x - h| > \sqrt{n}$ which can be written as $x - h < - \sqrt{n}$ or $x - h > \sqrt{n}$
- The final solution is $x < h - \sqrt{n}$ or $x > h + \sqrt{n}$

It is easiest to sketch the graph of a quadratic when it has a positive $x^{2}$ term, so rearrange first if necessary
Use your GDC to help select the correct region(s) for the inequality
Some makes/models of GDC may have the ability to solve inequalities directly
- However unconventional notation may be used to display the answer (e.g. $6 > x > 3$ rather than $3 < x < 6$ )
- The safest method is to always sketch the graph

Find the set of values which satisfy $3 x^{2} + 2 x - 6 > x^{2} + 4 x - 2$ .

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