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- 1. Quadratic Inequalities
- 2. 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
- 3. Guide Questions How 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.
- 4. 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
- 5. 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.
- 6. 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?
- 7. Plot the points corresponding to -3 and -4 on the number line. The three interval are: - ∞ < x < - 4, - 4 < x < - 3, - 3 < x < ∞. Test a number from each interval against the inequality. 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 > 0 49 – 49 + 12 > 0 12 > 0 (true) x2 + 7x + 12 > 0 (-3.6)2 + 7(-3.6) + 12 > 0 12.96 – 25.2 + 12 > 0 -0.24 > 0 (false) x2 + 7x + 12 > 0 (0)2 + 7(0) + 12 > 0 0 + 0 + 12 > 0 12 > 0 (true)
- 8. Also test whether the points x = - 3 and x = - 4 satisfy the equation. x2 + 7x + 12 > 0 (-3)2 + 7(-3) + 12 > 0 9 – 21 + 12 > 0 0 > 0 (false) x2 + 7x + 12 > 0 (-4)2 + 7(-4) + 12 > 0 16 – 28 + 12 > 0 0 > 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}.
- 9. 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

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