Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
137
Absolute Value Inequalities
Break these into two inequalities
Note: Problems 7.27–7.28 refer to the inequality .
7.27 Express the absolute value inequality as a compound inequality that does not
contain an absolute value expression and solve it for x.
Rewrite the absolute value inequality as the compound inequality
–b ≤ x + a ≤ b.
Isolate x between the inequality symbols.
Note: Problems 7.27–7.28 refer to the inequality .
7.28 Graph the solution to the inequality.
According to Problem 7.27, the solution to the inequality is –3 ≤ x ≤ –1. Plot
both boundaries using closed points, because x = –3 and x = –1 are solutions
to the inequality, and darken the portion of the number line between the
endpoints.
Figure 7-10: The graph of –3 ≤ x ≤ –1, the solution to the inequality .
7.29 Solve the inequality for x and graph the solution.
Express as a compound inequality that does not include an absolute
value expression, using the method described in Problem 7.27.
–5 ≤ 3x + 7 ≤ 5
Isolate x between the inequality symbols.
Graph the solution to the inequality using closed points to mark the
boundaries, as illustrated in Figure 7-11.
Take the
opposite of the
number that’s right
of the inequality
symbol, follow it up
with a copy of that
symbol, and after
that write the
original inequality
without the
absolute value
bars.
You can
only do this
“inequality
into a compound
inequality”
switcheroo if two
things are true:
(1) it’s an absolute
value inequality;
and (2) the sign is
either < or ≤. If the
sign is > or ≥, use
the technique
described in
Problems 7.31–
7.33.
Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
138
Figure 7-11: The graph of , the solution to the inequality .
7.30 Solve the inequality for x and graph the solution.
Before you express the absolute value inequality as a compound inequality,
isolate the absolute value expression left of the equal sign.
Express the absolute value inequality as a compound inequality and solve.
Isolating x requires you to divide by a negative number, so reverse the inequality
symbols.
The compound inequalities 3 > x > 0 and 0 < x < 3 are equivalent; the graph is
illustrated in Figure 7-12.
Figure 7-12: The graph of 0 < x < 3, the solution to the inequality .
Note: Problems 7.31–7.32 refer to the inequality .
7.31 Express the absolute value inequality as two inequalities that do not include
absolute value expressions and solve both for x.
Whereas absolute value expressions that contain either the < or ≤ symbol can
be rewritten as compound inequality statements, absolute value expressions
containing either the > or ≥ symbol cannot. Instead, express the inequality
as “x + a > b or x + a < –b.”
x – 4 > 2 or x – 4 < –2
Solve the inequalities.
Important
note here:
The absolute
values always
need to be left of
the inequality for
this method to work.
If they’re not, ip-
op the sides of the
inequality to move
it left where it
belongs. When you
do, don’t forget
to reverse the
inequality
sign.
To get the rst
inequality, drop the
absolute value bars.
To get the second,
drop the bars, reverse
the inequality symbol,
and take the opposite
of the constant on
the right side of the
original inequality.
Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
139
The solution to the inequality is x < 2 or x > 6. Therefore, any real number less
than (but not including) 2 or greater than (but not including) 6 makes the
inequality true.
Note: Problems 7.31–7.32 refer to the inequality .
7.32 Graph the inequality.
According to Problem 7.31, the solution to the inequality is x < 2 or x > 6. The
graph of the solution consists of the graphs of x < 2 and x > 6 on the same
number line. Plot x = 2 and x = 6 using open points, as illustrated in Figure 7-13;
they are not valid solutions to the inequality.
Figure 7-13: The graph of x < 2 or x > 6, the solution to the inequality .
7.33 Solve the inequality for x and graph the solution.
Isolate the absolute value expression left of the inequality symbol.
Use the method described in Problem 7.32 to express the absolute value
statement as two distinct inequalities that do not include absolute value
expressions and solve them.
The solution to the inequality, or , is graphed in Figure 7-14.
Figure 7-14: The solution to the inequality is or .
The word
“or” is impor-
tant. There’s no
number that’s both
greater than 6 AND less
than 2, so a solution of
“x < 2 AND x > 6”
doesn’t make
sense.
So the
graph is two
arrows shooting
different
directions, usually
with some space
between them—
much different
than the graphs
of absolute value
inequalities that
contain < or ≤, which
look like single
segments with
two endpoints.
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