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.

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