Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
484
Note: Problems 21.41–21.44 refer to the inequality .
21.42 Identify the critical numbers of the rational expression generated by Problem
21.41.
Factor the quadratic in the numerator.
Set each of the factors in the numerator and denominator equal to zero and
solve the resulting equations to identify the critical numbers.
The critical numbers of the rational expression are x = –3, , x = 0, and
x = 3.
Multiply
both sides of
the inequality by
–1. That changes
–2x
2
into 2x
2
, which
is slightly easier to
factor. Don’t forget to
reverse the inequality
sign because you’re
multiplying both sides
of the inequality
by a negative
number.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
485
Note: Problems 21.41–21.44 refer to the inequality .
21.43 Solve the inequality.
The critical numbers identified in Problem 21.42 (x = –3, , x = 0, and
x = 3), split the number line into five intervals, as illustrated by Figure 21-7.
Figure 21-7: The critical numbers split the number line into five distinct intervals:
x < –3, ,
3
x 0
2
- < <
, 0 < x < 3, and x > 3.
Choose one test point from each interval (such as x = –4, x = –2, x = –1, x = 1,
and x = 4) to identify the intervals that represent solutions to the inequality.
The solution to the inequality is x < –3 or or x > 3.
All the dots should
be open, because the
inequality sign is <.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
486
Note: Problems 21.41–21.44 refer to the inequality .
21.44 Graph the inequality.
Darken the intervals identified by Problem 21.43 as the solutions to the
inequality: x < –3 or or x > 3. The graph is presented in Figure 21-8.
Figure 21-8: The graph of .
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