Pre-Calculus Polynomial and Rational Inequalities
Solve the inequality algebraically. Use your graphing utility to check your solutions. x x 1 20 Factored but not equaled to zero. You must fix this. x x 1 20 x x 20 2
x x 20 0 2
x 4 x 5 0 x 4 x 5 These are our critical points. -5
4
Check points in the regions created by the critical points and check the critical points. For instance‌ -6
0 -5
7 4
Using the inequality “>” compare Y1 and Y2. For -6, 30 > 20 is true. Shade. For -5, 20 > 20 is not true. Open circle For 0, 0 > 20 is not true. Do not shade. For 4, 20 > 20 is not true. Open circle. For 7, 56 > 20 is true. Shade.
Final solution, in interval notation: ,5 4,
Solve the inequality algebraically. Use your graphing utility to check your solutions.
x 5
2
x 4 2
0
First, the critical point from the denominator. x2 4 0 x2 4 x 2
Now solve the expression as if it equals.
x 5
2
0
x 4 2
x 5
2
0
x 5 0 x 5
-2, 2, and -5 are your critical points.
-5
-2
2
Check points in the regions created by the critical points and check the critical points. For instance‌ -4
-6 -5
4
0 -2
2
Using the inequality “≥” compare Y1 and Y2. For -6, .03125 ≥ 0 is true. Shade. For -5, 0 ≥ 0 is true. Closed circle For -4, .08333 ≥ 0 is true. Shade. For -2, ERROR ≥ 0, is not true. Open circle. For 0, -6.25 ≥ 0 is not true. Do not shade. For 2, ERROR ≥ 0 is not true. Open circle. For 4, 6.75 ≥ 0 is true. Shade. Final solution, in interval notation: , 2 2,