2. Solving Nonlinear Inequalities Section 2-7

3. 2 Solution to Inequality Equation One solution Inequality Infinite Solutions

4. 3 Graphing Inequalities 0 [, ] – number is included (, ) – number is not included (, ) – always used with,

5. 4 Solution Set The solution set of an inequality is the set of all solutions. Study the graph of the solution set of x 2 + 3x  4  0. The solution set is {x |  4  x  1}. The values of x for which equality holds are part of the solution set. These values can be found by solving the quadratic equation associated with the inequality. x 2 + 3x  4 = 0 Solve the associated equation. (x + 4)(x  1) = 0 Factor the trinomial. x =  4 or x = 1 Solutions of the equation -2012-6-5- 4- 4-3 ][

6. 5 Vocabulary The Critical Numbers of any rational expression inequality are the zeros and undefined numbers. Critical Numbers for a quadratic inequality are the roots. These numbers will be used to establish the test intervals over which we will solve the inequalities. Zeros are where the numerator will be zero Undefined Numbers are where the denominator will be zero.

7. 6 Solve: x 2 - 8x – 33 > 0 x 2 - 8x – 33 > 0 (x - 11) (x + 3) > 0 x = 11 or x = -3 (These are critical numbers) Test 3 areas x 11 See next slide.

8. 7 Solve: x 2 - 8x – 33 > 0 x = 11 or x = -3 -3 11 Test (-5) (-5) 2 - 8(-5) – 33 > 0 25 + 40 – 33 > 0 65 – 33 > 0 32 > 0 YES Test (0) 0 2 - 8(0) – 33 > 0 -33 > 0 NO Test (15) 15 2 –8(15) – 33 > 0 225 – 120 – 33 > 0 105 – 33 > 0 72 > 0 YES Solutions: (-∞, -3) U (11, ∞)

9. 8 Solve and graph :

10. 9 Unusual Solution Sets There are 4 cases of possible unusual solutions: 1. The solution can be all real numbers (- ,  ) 2. The solution may be a single number {3} 3. The solution may be the Null Set, (No solution) 4. The solution may be all numbers except one, like (- , 3) U (3,  )

11. 10 Homework WS 4-5 Quiz next class