The Humongous Book of Algebra Problems

Chapter Seven — Linear Inequalities

The Humongous Book of Algebra Problems

132

Graphing Inequalities in One Variable

Shoot arrows into number lines

7.13 What characteristics of an inequality statement determine whether the point

or points plotted on its graph are open or closed?

Points on the graph of a linear inequality are classified according to the

adjacent inequality symbol. If that symbol allows for the possibility of equality

(that is, the symbol is ≤ or ≥), plot the value using a closed point. Alternatively,

if the symbol indicates strict inequality (the symbol is < or >), an open point

should be used to indicate that the value is not a solution to the inequality.

7.14 Compare and contrast the graphs of linear equations and linear inequalities in

one variable.

Because they each contain one variable, both are plotted on a number line

(rather than a coordinate plane, which is used when statements are written in

terms of two variables). The graphs of linear equations consist of a single point

on the number line, whereas the solutions of linear inequalities are intervals

consisting of infinitely many values.

7.15 Graph the inequality: x > 3.

Plot the value x = 3 on a number line using an open point, as explained in

Problem 7.13. Darken the portion of the number line that is right of x = 3,

as illustrated by Figure 7.1, as any value in that interval makes the inequality

true.

Figure 7-1: The graph of x > 3 has an open point at x = 3 because 3 is not a valid

solution to the inequality.

7.16 Graph the inequality x ≤ –1.

Plot the value x = –1 on a number line using a closed point, as x = –1 is one of

the solutions to the inequality. Darken the portion of the number line that

is left of –1 to identify the other solutions of the inequality, as illustrated in

Figure 7-2.

Figure 7-2: The graph of x ≤ –1 consists of –1 and all of the real numbers less than –1.

“Open”

points are

not included

in the graph

and look like

hollow dots.

“Closed” points are

included on the

graph and look

like solid

dots.

If you

say, “I can’t

run faster than

9 miles an hour,”

and you mean that

9 mph is an impossible

goal, your speeds are

x ≤ 9, and the graph

would have an open

dot. If you mean you

CAN run 9 mph but

no more, then use

a closed dot to

graph x ≤ 9.

Intervals

are seg-

ments of the

number line, so

the solution to a

linear inequality is

something like x > –4

(any real number

greater than –4 is

a solution), and the

solution to a linear

equation would

be a single

number like

x = –4.

The inequality > points right, so shade everything

right of x = 3 on the number line. This shortcut works only

when x is on the left side of the inequality.

Chapter Seven — Linear Inequalities

The Humongous Book of Algebra Problems

133

7.17 Solve the inequality 3x + 20 ≥ 8 for x and graph the solution.

Solve the inequality by isolating x left of the inequality sign.

Plot x = –4 on the number line using a closed point and darken the portion of

the number line greater than –4, as illustrated by Figure 7-3.

Figure 7-3: The graph of x ≥ –4, the solution to the inequality 3x + 20 ≥ 8.

7.18 Solve the inequality for x and graph the solution.

Multiply both sides of the inequality by 4 to eliminate the fraction and then

isolate x left of the inequality symbol.

Plot on a number line using an open point and darken the portion of

the number line that is less than , as illustrated by Figure 7-4.

Figure 7-4: The graph of x < , the solution to the inequality (5x – 3) < 2.

If the

inequality symbol

is either < or >, use

an open dot on the

graph. If it’s ≤ or ≥,

use a closed dot.

It’s easier to

graph this fraction

if you write it as a

mixed number. Use the

formula from Problem

2.7: 5 divides into 11

twice with a remainder

of 1, so

.