Quadratic Inequalities

Activity 1: What Makes Me True?

Give the solution/s of each of the following mathematical sentences.1. x + 5 > 8

2. r – 3 = 10

3. 2s + 7 ≥ 21

4. 3t – 2 ≤ 13

5. 12 – 5m = - 8

Guide QuestionsHow did you find the solution/s of each

mathematical statements?

What mathematical concepts or principles did you apply to come up with the solution/s?

Which mathematical sentences has only one solution? More than one solution? Describe these sentences.

Activity 2: Which are Not Quadratic Equations?

x2 + 9z + 20 = 0

2r2 < 21 - 9t 2x2 + 2 = 10x

r2 + 10r ≤ - 16 m2 = 6m - 7 4x2 – 25 = 0

15 – 6h2 = 10 3w2 + 12w ≥ 0

2s2 + 7s + 5 > 0

Definition Is an inequality that contains a polynomial of degree

2 and can be written in any of the following forms.

ax2 + bx + c > 0 ax2 + bx + c ≥ 0

ax2 + bx + c < 0 ax2 + bx + c ≤ 0

where a, b, and c are real numbers and a ≠ 0.

To solve a quadratic inequality, find the roots of its corresponding equality.

Find the solution set of x2 + 7x + 12 > 0.The corresponding equality of x2 + 7x + 12 > 0 is x2 + 7x + 12 = 0.

Solve x2 + 7x + 12 = 0.(x + 3)(x + 4) = 0 Why?x + 3 = 0 & x + 4 = 0 Why?x = - 3 & x = - 4 Why?

Plot the points corresponding to -3 and -4 on the number line.

For - ∞ < x < - 4,Let x = - 7

For – 4 < x < - 3, Let x = 3.6

For – 3 < x < ∞,Let x = 0

x2 + 7x + 12 > 0(-7)2 + 7(-7) + 12 > 049 – 49 + 12 > 012 > 0 (true)

x2 + 7x + 12 > 0(-3.6)2 + 7(-3.6) + 12 > 012.96 – 25.2 + 12 > 0-0.24 > 0 (false)

x2 + 7x + 12 > 0(0)2 + 7(0) + 12 > 00 + 0 + 12 > 012 > 0 (true)

The three interval are: - ∞ < x < - 4, - 4 < x < - 3, - 3 < x < ∞.

Test a number from each interval against the inequality.

Also test whether the points x = - 3 and x = - 4 satisfy the equation.

x2 + 7x + 12 > 0(-3)2 + 7(-3) + 12 > 09 – 21 + 12 > 00 > 0 (false)

x2 + 7x + 12 > 0(-4)2 + 7(-4) + 12 > 016 – 28 + 12 > 00 > 0 (false)

Therefore, the inequality is true for any value of x in the interval - ∞ < x < - 4 or - 3 < x < ∞, and these intervals exclude – 3 and – 4. The solution set of the inequality is {x:x < - 4 or x > - 3}.

Quadratic Inequalities In Two Variables

There are quadratic inequalities that involves two variables. These inequalities can be written in any of the following forms.

y > ax2 + bx + c y ≥ ax2 + bx + c

y < ax2 + bx + c y ≤ ax2 + bx + c