8.3 Quadratic Inequalities

1. 8.3 Quadratic Inequalities

2. A quadratic can be combined with inequalities If ax2 + bx + c > 0 we are looking for values of x that makes the quadratic greater than zero If ax2 + bx + c < 0, looking where the quadratic is less than zero • Ex 1) Solve –2x2 + 4x + 9 ≥ 0 with the aid of a graph. • *graph on TV* Find intercepts –1.3 and 3.3 • Where is quadratic greater than or equal to 0? (above x-axis) [–1.3, 3.3]

3. We can also solve them algebraically without using a picture. Ex 2) Solve –x2 – 4x + 5 ≤ 0 algebraically x2 + 4x – 5 ≥ 0 (x + 5)(x – 1) ≥ 0 x = –5, 1 • closed circles • want (+) values • Choose “test points” in each interval –5 1 (–)(–) (+) (+)(–) (–) (+)(+) (+) Is the value (+) or (–)? (don’t care about actual #) (–∞, –5]  [1, ∞)

4. They don’t always have to be integer answers. • Ex 3) Solve 6x2 + 10x – 5 ≤ 0 algebraically • can’t factor •  quadratic formula! • = 0.40 or –2.07 Want (–) –2.07 0.40 0 –5 (–) 1 6 + 10 – 5 (+) –3 54 – 30 – 5 (+) [–2.07, 0.40]

5. Ex 4) The NCAA Men’s Basketball Rulebook states the height of the basket on a regulation court must be 10 ft. For a particular throw the height s of a basketball at any time t, in seconds, can be determined by the equation s = –16t2 + 40t + 6. Determine for what values of t, to the nearest tenth of a second, the ball is higher than the basket. Express in interval notation. –16t2 + 40t + 6 > 10 –16t2 + 40t – 4 > 0 • = 0.1 or 2.4 Want (+) 0.1 2.4 0 – 4 (–) 1 –16 + 40 – 4 (+) 3 –144 + 120 – 4 (–) (0.1, 2.4)

6. We can also graph quadratic inequalities *Remember < or > dotted line y < shade below ≤ or ≥ solid line y > shade above Ex 5) Graph in the coordinate plane 4 – y ≥ 5x2 – 2x –y ≥ 5x2 – 2x – 4 y ≤ –5x2 + 2x + 4 Solid line A of S: x = Vertex: y-int: shade below (0, 4) *You can also use a test point to confirm such as (0, 0) 4 – 0 ≥ 5(0)2 – 2(0) 4 ≥ 0 Yes  include (0,0)

7. Homework #803 Pg 404 #1, 2, 5, 12, 13, 18, 19, 20, 22, 27, 31, 38, 42, 47