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“In third grade, I cheated on my history exam. In fourth grade, I stole my uncle Max's toupee and I glued it on my face when I was Moses in my Hebrew School play. In fifth grade, I knocked my sister Edie down the stairs and I blamed it on the dog....” 3.5 Polynomial and Rational Inequalities Solving Equations Let’s review solving the following equation: 2x x 6 0 2 f (x) 2x 2 x 6 (-1.5,0) (2,0) 2x 2 x 6 0 *A polynomial function can only change signs at its zeros (x-ints).* Solving Inequalities Solve the following inequality without graphing: 6x 6 5x 2 Solving Polynomial Inequalities What steps did we use to solve 6x 2 6 5x ? 6x 2 5x 6 0 1) Get everything on the left side to set equal to zero. (2x 3)(3x 2) 0 Zeros : (2 3,0),(3 2,0) 2) Find the zeros of f (x) by factoring. 32 2 3 f (1) f (0) f (2) 3) Use zeros to separate real number line into test intervals. 4) Pick value from each testinterval to evaluate sign of f (x). (,2 3][3 2,) 5) Write out solution region in interval notation. Solving Inequalities Solve the following inequality without graphing: x 4 x 4 x 16 3 2 Solving Inequalities Solve the following inequality without graphing: x 3 3x 2 Solving Inequalities Solve the following inequalities without graphing: 1) x 3 2x 2 3x 0 2) x 4 13x 36 0 2 Solving Rational Equations Let’s review solving the following equation: 4x 5 0 x2 4x 5 f (x) x2 4x 5 0 x2 (-1.25,0) *A rational function can change signs at its zeros (x-ints) AND where f is undefined.* Solving Inequalities Solve the following inequality without graphing: 3 3x 2 2x 5 Solving Rational Inequalities What steps did we use to solve rational inequalities? 3 3x 2 0 2x 5 1) Get everything on the left side and zero on the other. 7x 7 0 Zero : (-1,0) DNE : x 5 2 2x 5 2) Find the zeros AND where f (x) DNE by finding common denominator. 1 5 2 f (3) f (0) f (1) 3) Use these to separate real number line into test intervals. 4) Pick value test interval to evaluate sign of f (x). from each (5 2,1] 5) Write out solution region in interval notation. Solving Inequalities Solve the following inequality without graphing: 5 3 x 3 x 1 Solving Inequalities Solve the following inequality without graphing: 15 x 8 x Practice Problem Solve the following inequality without graphing: x4 1 2x 4 3.5 Polynomial and Rational Inequalities Homework #3: Page 217 #11, 17, 23, 33 – 37 Odd, 43, 45, 49