2. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Aim: How do we solve quadratic inequalities? Do Now: What are the roots for y = x 2 - 2x - 3?

3. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Graph y = x 2 - 2x - 3 where the parabola crosses the x-axis Finding the roots/zeroes: Graphically: -1,03,0 x-axis y = 0 represents the x-axis and the solution to quadratic x 2 - 2x - 3 = 0 is found at the intersection of the parabola and x-axis 0 = x 2 - 2x - 3 0 = (x - 3)(x + 1) x = 3 and x = -1 factor and solve for x. Algebraically: x-intercepts

4. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Graphing a Linear Inequality Graph the inequality y - 2x > 2y - 2x > 2 1. Convert to standard form +2x +2x y > 2 + 2x 2. Create Table of Values y = 2 + 2x 3. Shade the region above the line. xy2 + 2x 0 1 2 2 4 6 2 + 2(0) 2 + 2(1) 2 + 2(2) 4. Check the solution by choosing a point in the shaded region to see if it satisfies the inequality (-2,4) 4 - 2(-2) > 2 4 - (-4) > 2 8 > 2 y - 2x > 2 Note: the line is now solid.

5. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Graphing a Linear Inequality An inequality may contain one of these four symbols:, >, or <. y < mx + by > mx + b The boundary line is part of the solution. It is drawn as a SOLID line. The boundary line is not part of the solution. It is drawn as a DASHED line. y < mx + by > mx + b

6. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Aim: How do we solve quadratic inequalities? Do Now: Graph: y – 3x < 3 Solve: y – 3x 2 = 9x – 12

7. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. y > x 2 - 2x - 3y < x 2 - 2x - 3 Quadratic Inequalities - Graphically y > x 2 - 2x - 3 & y < x 2 - 2x - 3 -1,03,0 -1,03,0 -1 < x < 3x 3 > Shaded inside the curve< Shaded outside the curve The values of x found within the shaded regions. 0 > x 2 - 2x - 3 0 < x 2 - 2x - 3 What values of x satisfy these inequalities when y = 0 (x, y)

8. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Graph y ≥ x 2 - 1 or x 2 - 1 ≤ y Because y is greater than or equal to (≥) x 2 - 1, the parabola is shaded inside the curve and includes the curve itself Graphically: -1 ≤ x ≤ 1 What values of x satisfy the quadratic inequality when y = 0? x-axis (x 2 – 1 = 0) -1,01,0 roots

9. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Exceptions - 1 What values of x satisfy the quadratic inequality 0 > x 2 - 4x + 4? y < x 2 - 4x + 4 Solution: {x| x = 2} y > x 2 - 4x + 4 x = 2 root/zero What values of x satisfy the quadratic inequality 0 < x 2 - 4x + 4? y > x 2 - 4x + 4 Solution:  = (x - 2)(x - 2) 0 = Quadratic Inequalities that have roots that are equal (2,0) 0 y > x 2 - 4x + 4 0

10. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Exceptions - 2 Quadratic Inequalities that have no roots. What values of x satisfy the quadratic inequality 0 > x 2 + 1? Solution: {x| x =  } What values of x satisfy the quadratic inequality 0 < x 2 + 1? y > x 2 + 1 Solution: {x| x =  } (0, 1) y < x 2 + 1 (0, 1)

11. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. General Solutions of Quadratic Inequalities where a > 0 and r 1 < r 2 (r 1 and r 2 are the unequal roots) Quadratic Inequality Solution Interval Graph of Solution ax 2 + bx + c < 0r 1 < x < r 2 ax 2 + bx + c < 0r 1 < x < r 2 ax 2 + bx + c > 0r 1 < x or x > r 2 ax 2 + bx + c > 0r 1 < x or x > r 2 r1r1 r2r2 r1r1 r2r2 r1r1 r2r2 r1r1 r2r2

12. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Critical Numbers & Test Intervals x 2 – 2x – 3 < 0 (-1,0)(3,0) roots, or zeros Critical Numbers for testing the inequality (x + 1)(x – 3) = 0 x = -1 and x = 3 are the roots or the zeros that create 3 test intervals (- , -1) (-1, 3) (3,  ) Test Interval Representative x-value Value of Polynomial x = -3 (-3) 2 – 2(-3) – 3 = 12 x = 0 (0) 2 – 2(0) – 3 = -3 (- , -1) (-1, 3) (3,  ) x = 5 (5) 2 – 2(5) – 3 = 12 (-1, 3) Is this value < 0? No Yes No

13. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Model Problems Test Interval Representative x-value Value of Polynomial Solve algebraically and Graph: y < x 2 – 12x + 27 0 < x 2 – 12x + 27

14. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Model Problems Graph the solution set for x 2 – 2 > -x – 3

15. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Regents Question Which graph best represents the inequality y + 6 > x 2 – x? 1) 2) 3) 4)

16. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Model Problems Graph the solution set for 2(x – 2)(x + 3) < (x – 2)(x + 3)

17. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Because 0 ≥ x 2 - 1, x 2 - 1 must be a negative number or 0 Solve 0 ≥ x 2 - 1 algebraically -1 ≤ x ≤ 1 Algebraically: 0 ≥ x 2 - 1 0 ≥ (x - 1)(x + 1) x ≥ 1 and x ≥ -1 ? 0 ≥ (x - 1)0 ≥ (x + 1) a negative number is the product of a positive & negative #  one of the factors must be positive and the other negative If ab 0, or a > 0 and b < 0. -1,01,0 roots

18. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Solve 0 ≥ x 2 - 1 algebraically (con’t) x 2 - 1 ≤ 0 (x - 1)(x + 1) ≤ 0 (x - 1) ≥ 0(x + 1) ≤ 0(x - 1) ≤ 0(x + 1) ≥ 0 x ≥ 1x ≤ -1x ≤ 1x ≥ -1 and x CANNOT be a number less than or equal to -1 and greater than or equal to 1. EXTRANEOUS Set quadratic = 0 Factor What values of x can satisfy both inequalities for each set? “What values of x are less than or equal to 1 and greater than or equal to -1?” -1 ≤ x ≤ 1 KEY WORD - “and”

19. Aim: Quadratic Inequalities Course: Adv. Alg. & Trig. Aim: How do we solve quadratic inequalities? Do Now: Graph the inequality y ≥ x 2 – 1