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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 cant
run faster than
9 miles an hour,
and you mean that
9 mph is an impossible
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
.

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