Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
479
Solving Rational Inequalities
Critical numbers, test points, and shading
Note: Problems 21.31–21.33 refer to the inequality .
21.31 Identify the critical numbers of the rational expression.
The critical numbers of an expression are the values for which the expression
equals zero or is undefined. A rational expression equals zero when its
numerator equals zero and is undefined when its denominator equals zero. Set
the numerator and denominator of the fraction equal to zero and solve the
resulting equations.
The critical numbers of the rational expression are x = –2 and x = 3.
Note: Problems 21.31–21.33 refer to the inequality .
21.32 Identify the intervals of the real number line that represent solutions to the
inequality.
The critical numbers x = –2 and x = 3 (calculated in Problem 21.31) split the real
number line into three intervals, as illustrated by Figure 21-1.
Figure 21-1: The critical numbers x = –2 and x = 3 split the real number line into three
intervals: x < –2, –2 < x < 3, and x > 3. Use open points to mark the critical numbers,
because the inequality sign “>” does not allow for the possibility of equality.
Choose one test value from within each interval (such as x = –5 from x < –2,
x = 0 from –2 < x < 3, and x = 5 from x > 3) and substitute those values into the
inequality.
The solution to the inequality is x < –2 or x > 3.
The symbols
< and > always
mean “do not include
the boundaries,”
whether you use
hollow dots on a
number line or
dotted lines on
a coordinate
plane.
The test
points from these
intervals produced
true statements, so
that’s why they make
up the solution. Use
the word “or” between
the solution intervals,
because there aren’t
any numbers that
are fewer than
–2 and greater
than 3.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
480
Note: Problems 21.31–21.33 refer to the inequality .
21.33 Graph the inequality.
Darken the segments of the number line identified by Problem 21.32 as
solutions to the inequality: x < –2 and x > 3. The graph is presented in
Figure 21-2.
Figure 21-2: The graph of .
Note: Problems 21.34–21.36 refer to the inequality .
21.34 Identify the critical numbers of the expression.
Factor the numerator.
The critical numbers of a rational expression are the x-values that cause either
the numerator or denominator to equal zero. Set each of the factors equal to
zero and solve for x.
Note: Problems 21.34–21.36 refer to the inequality .
21.35 Solve the inequality.
According to Problem 21.34, the critical numbers of the rational expression are
x = –4, , and x = 5. Plot the values on a number line, noting that x = 5 is
represented by an open point, whereas the other two critical numbers are
plotted as closed points. As illustrated by Figure 21-3, the critical numbers
divide the number line into four intervals.
Normally
≤ and ≥
mean solid dots,
but x = 5 makes
the denominator
zero, which is not
allowed. Anything
that makes the
denominator zero
becomes an open dot
on the number line
no matter what
the inequality
symbol is.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
481
Figure 21-3: The critical numbers separate the number line into four intervals:
x ≤ –4, , , and x > 5.
Substitute a test value from each interval (such as x = –6, x = 0, x = 3, and x = 8)
into the inequality to identify the solutions.
The solution to the inequality is or x > 5.
Note: Problems 21.34–21.36 refer to the inequality .
21.36 Graph the inequality.
Darken the intervals of the number line identified by Problem 21.35 as solutions
to the inequality: and x > 5. The graph is presented in Figure 21-4.
Figure 21-4: The graph of .
The inequality
symbols attached
to x = 5 are ‹ and ›,
instead of ≤ and ≤,
because x = 5 is an
open dot on the
graph.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
482
Note: Problems 21.37–21.40 refer to the inequality .
21.37 Write the left side of the inequality as a single rational expression.
Express the left side of the inequality using the least common denominator
2(x – 1).
Note: Problems 21.37–21.40 refer to the inequality .
21.38 Identify the critical numbers of the rational expression generated in Problem
21.37.
Factor the numerator.
Set each of the factors equal to zero and solve the resulting equations.
As illustrated by Figure 21-5, the critical numbers x = –2, x = 1, and x = 3 split the
number line into four intervals: x ≤ –2, –2 ≤ x < 1, 1 < x ≤ 3, and x ≥ 3.
Figure 21-5: The critical numbers x = –2, x = 1, and x = 3 split the number line into
four intervals. Plot x = 1 as an open point because it causes the denominator to equal zero,
making the rational expression undefined.
When you
solve a rational
inequality, you want
the right side of the
statement to be zero.
It’s not like an equation,
where you add
to both sides
and cross multiply.
If you don’t know
how to do this, look at
Problems 20.9–20.19.
Except
the factor
2 from the
denominator,
because 2 by
itself can’t
equal 0.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
483
Note: Problems 21.37–21.40 refer to the inequality .
21.39 Solve the inequality.
Substitute a test value from each of the intervals identified by Problem 21.38
(such as x = –3, x = 0, x = 2, and x = 4) into the inequality to identify the
solution.
The solution to the inequality is x ≤ –2 or 1 < x ≤ 3.
Note: Problems 21.37–21.40 refer to the inequality .
21.40 Graph the inequality.
Darken the segments of the number line identified by Problem 21.39 as
solutions to the inequality: x ≤ –2 and 1 < x ≤ 3. The graph is presented in
Figure 21-6.
Figure 21-6: The graph of the inequality .
Note: Problems 21.41–21.44 refer to the inequality .
21.41 Combine the rational expressions into a single rational expression.
Subtract from both sides of the inequality.
Combine the fractions using the least common denominator x(x + 3).
The test
values are
plugged into the
factored version
of the fraction from
Problem 21.38, but you
could also use the
original inequality.
So that the
right side of the
inequality equals
zero.
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