Chapter Seven — Linear Inequalities

The Humongous Book of Algebra Problems

142

7.38 Express the solution to the inequality in set notation.

Multiply both sides of the inequality by –4 to isolate the absolute value

expression. Recall that multiplying by a negative number reverses the inequality

symbol.

The left side of the inequality consists of an absolute value statement, so no

matter what value of x is substituted into , its value is nonnegative.

The inequality stipulates that the absolute value expression is less than or equal

to –20, but a nonnegative number cannot be less than a negative number, so

there are no real number solutions to the inequality. Because the set of solutions

is empty, the solution set is the empty, or null, set: .

7.39 Express the solution to the inequality in set notation.

Like Problem 7.38, the absolute value expression left of the inequality symbol

is nonnegative for any real number x, so all real numbers are solutions to the

inequality. In set notation, the solution is {x : x is a real number}; in other words,

if x is a real number, then x is a valid solution to the inequality.

Graphing Inequalities in Two Variables

Lines that give off shade in the coordinate plane

7.40 Graph the inequality y ≤ –3x – 1.

The linear inequality is written in terms of two variables, x and y, so it must

be graphed on a system with two axes, the coordinate plane. Begin by

graphing y = –3x – 1, an equation in slope-intercept form, using the technique

described in Problems 6.21–6.26. However, use a dotted line rather than a

solid line because the points on the line are not solutions to the inequality. The

dotted line separates the coordinate plane into two distinct regions, labeled

A and B in Figure 7-16.

If you plug in

x = 6, you get

.

Any other x-value

produces a positive

number.

The null

set symbol

is NOT a zero,

because zero is a

real number. The

solution set for the

equation x + 5 = 5

is {0}, because x = 0

makes that equation

true and is, there-

fore, a solution. A null

solution set means

there are NO

solutions.

If the left

side is always

greater than or

equal to 0, then it’s

always larger than

the right side, which

is –1.

RULE OF THUMB: When you

see < or > and you’re graphing on a

number line, you use open dots. When

you see < or > and you’re graphing on

a coordinate plane, use a dotted line.

Similarly, ≤ and ≥ indicate solid dots on

the number line and solid lines on the

coordinate plane.

Chapter Seven — Linear Inequalities

The Humongous Book of Algebra Problems

143

Figure 7-16: The line y = –3x – 1 has y-intercept (0,–1) and slope –3 but does not

represent the graph of the inequality y ≤ –3x – 1. In fact, because the line

is dotted, it is not part of the graph at all. Rather, it splits the coordinate

plane into two regions, here labeled A and B.

Either region A or region B represents the solution to the inequality. Choose

one “test” point, from either of the regions, and substitute it into the inequality.

If that test point makes the inequality true, then the region to which it belongs

is the solution. If it makes the inequality false, then the region not containing

the test point is the solution.

For instance, the point (–2,0) falls within region A in Figure 7-16. Substituting

x = –2 and y = 0 into the inequality y ≤ –3x – 1 produces a true statement.

Because (–2,0) makes the inequality true, region A is the solution to the

inequality. Lightly shade that region on the coordinate plane to complete the

graph, as illustrated by Figure 7-17.

You can

pick any test

point you want, as

long as it’s clearly

in one of the two

regions. Choosing test

points along the line

(whether it’s solid

or dotted) doesn’t

help at all.

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