Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
135
Compound Inequalities
Two inequalities for the price of one
Note: Problems 7.21–7.22 refer to the compound inequality –4 < x ≤ 5.
7.21 Express the compound inequality as two inequalities in terms of x.
The compound inequality –4 < x ≤ 5 consists of the inequalities x > –4 and x ≤ 5.
Therefore, every real number x that satisfies the inequality –4 < x ≤ 5 is between
–4 and 5, including x = 5 but excluding x = –4.
Note: Problems 7.21–7.22 refer to the compound inequality –4 < x ≤ 5.
7.22 Graph the inequality.
A compound inequality describes a segment of the number line defined by two
endpoints, in this case x = –4 and x = 5. Plot the endpoints as open or closed
points depending upon the adjacent inequality symbol.
The < symbol adjacent to the boundary –4 prohibits equality, so plot it using an
open point. The ≤ symbol next to x = 5, on the other hand, indicates that 5 is a
solution to the inequality, so plot it using a closed point.
Complete the graph by shading the portion of the number line lying between
the endpoints, as illustrated by Figure 7-7.
Figure 7-7: The graph of the inequality –4 < x ≤ 5 includes x = 5 but excludes x = –4.
Note: Problems 7.23–7.24 refer to the compound inequality –2 < x – 4 < 1.
7.23 Solve the inequality for x.
Isolate x between the inequality symbols by adding 4 to each expression of the
inequality.
Note: Problems 7.23–7.24 refer to the compound inequality –2 < x – 4 < 1.
7.24 Graph the inequality.
According to Problem 7.23, the solution to the inequality is 2 < x < 5. Use open
points to plot the endpoints x = 2 and x = 5 and darken the portion of the
number line between the boundaries to complete the graph, as illustrated by
Figure 7-8.
Figure 7-8: The graph of 2 < x < 5, the solution to the inequality –2 < x – 4 < 1.
Just like
x is WRITTEN
between
–4 and 5 in
the inequality
–4 < x ≤ 5, all the
SOLUTIONS are
between those
boundaries as
well. The right
boundary is a
solution because of
the ≤ symbol, but
because of the <
symbol, the left
boundary is
not.
These rules
about when dots
are open and when
they’re closed are
consistent with Problem
7.13. Basically, if the
symbol includes “or equal
to,” use a solid dot.
Otherwise, use a
hollow one.
Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
136
Note: Problems 7.25–7.26 refer to the compound inequality .
7.25 Solve the inequality for x.
Eliminate by multiplying each of the expressions by 2.
Isolate x between the inequality symbols.
Note: Problems 7.25–7.26 refer to the compound inequality .
7.26 Graph the inequality.
Plot the boundary using a closed point (because the adjacent inequality
symbol is ≤) and plot using an open point (because the adjacent
inequality symbol is <). Darken the portion of the number line bounded by
those x-values, as illustrated by Figure 7-9.
Figure 7-9: The graph of , the solution to the inequality
.
Converting
into the
mixed number
makes it easier to
graph.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.