MA 3972-MA-Book May 8, 2018 13:46
Review of Precalculus 55- Entery=x^3 − 8 xinto your graphing calculator.
- Find the zeros ofy:x=−3, 0, and 3.
- Determine the intervals on whichy<0: (−∞,−3) and (0, 3).
- Check whether the endpoints satisfy the inequality. Since the inequality is strictly less
than 0, the endpoints are not included in the solution. - Write the solution to the inequality. The solution is (−∞,−3)∪(0, 3).
Example 3
Solve the inequalityx^3 − 9 x<0 algebraically.
- Write in standard form:x^3 − 9 x<0 is already in standard form.
- Factor the polynomial:x(x−3)(x+3)
- Find zeros:x(x−3)(x+3)=0 implies thatx=0,x=3, andx=−3.
- Determine the intervals:
(−∞,−3), (−3, 0), (0, 3), and (3,∞) − 3 03- Select anx-value and evaluate the polynomial:
SELECTED FACTOR FACTOR POLYNOMIAL
INTERVAL x-VALUE FACTORx (x+3) (x−3) x(x−3)(x+3)
(−∞,−3) − 5 −−− −
(−3, 0) − 1 −+− +
(0, 3) 1 ++− −
(3,∞)6+++ +Therefore, the intervals (−∞,−3) and (0, 3) makex(x−3)(x+3)<0.- Check the endpoints: Since the inequalityx^3 − 9 x<0 is strictly less than 0, none of
the endpointsx=−3, 0, and 3 are included in the solution. - Write the solution: The solution is (−∞,−3)∪(0, 3).
Solving Rational Inequalities
- Rewrite the given inequality so that all the terms are on the left and only zero is on the
right. - Find the least common denominator, and combine all the terms on the left intoa single
fraction. - Factor the numerator and the denominator, if possible.
- Find allx-values for which the numerator or the denominator is zero.
- Putting thesex-values on a number line, determine the test intervals.
- Select anx-value from each interval and substitute it in the fraction.
- Check theendpointsof each interval with the inequality.
- Write the solution to the inequality.