Page 1 of 2
Math 112
Section 2.7 Polynomials and Rational Inequalities
Steps to solve polynomial inequalities:
1. Put zero on one side.
2. Solve the equation (find zeroes). These are critical points
3. Make a number line and place critical numbers on number line. Test a number in each interval.
4. Pick interval that satisfies the inequality symbol. (If f(x) > 0, then we want the positive
intervals. If f(x) < 0, then we want the negative intervals.)
Example 2: 4 x 3  7 x 2  15 x
Example 1: ( x  1)( x  4)  0
Visual Solution of x3 + 6x2 – x – 30 < 0
10
5
-10
10
-5
-10
Visual Solution of x3 + 6x2 – x – 30 > 0
-15
-20
-25
-30
Example 3:
Steps to solve rational inequalities:
1. Find the values not defined in denominator. (critical
Visualnumbers)
Solution of x3 + 6x2 – x – 30 > 0
2. Put zero on one side.
3. Solve the related equality by multiplying by the LCD. These are critical numbers.
4. Make a number line and place critical numbers on number line. Test a number in each interval.
5. Pick interval that satisfies the inequality symbol.
Example 4: Find the interval that satisfies:
2
 0.
5 x
Visual Solution of x3 + 6x2 – x – 30 > 0
Example 5: Find the interval that satisfies:
Page 2 of 2
x 1 x  3

 0.
x  2 x 1
15
Visual Solution of
8x
0
x 4
Visual Solution of
8x
0
x 4
2
10
5
-10
10
-5
2
-10
-15
Example 6:
Example 7:
2
Flexl, Inc., determines that its total profit is given by the function P( x)  3x  630 x  6000 .
Flexl makes a profit for those nonnegative values of x for which P( x)  0 . Find the nonnegative
values of x for which Flexl makes a profit. Find the values of x for which Flexl loses money.