Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
140
7.34 Solve the inequality for x and graph the solution.
Isolate the absolute value expression by subtracting 1 from both sides of the
inequality.
To solve an absolute value inequality using the methods described in the
preceding examples, the absolute value expression must be left of the inequality
symbol.
Apply the technique described in Problem 7.31 to solve the inequality.
The solution to the inequality is x < –10 or x > –4 and is graphed in Figure 7-15.
Figure 7-15: The graph of x < –10 or x > –4, the solution to the inequality
.
Set Notation
A fancy way to write solutions
7.35 Express the solution to the equation in set notation.
Express the absolute value equation as two distinct equations that do not
contain absolute values and solve them for x.
The solution set of the inequality is .
When you
ip-op the sides
of an inequality, you
have to reverse the
inequality symbol.
This is an
absolute value
EQUATION, not
an inequality. If
you’re not sure how
to solve it, ip back
to Problem 4.33,
which is very
similar.
This is not a
coordinate. It’s
a list, or set, of
all the solutions
to the equation.
When there’s a xed
number of answers,
expressing them in
set notation just
means listing
them inside
braces.
Chapter Seven — Linear Inequalities
The Humongous Book of Algebra Problems
141
7.36 Express the solution to the inequality in set notation.
Express the absolute value inequality as a compound inequality that does not
contain an absolute value expression, as explained in Problems 7.27 and 7.29,
and solve the inequality.
The solution set of the inequality is written either as {x : –23 < x < –1} or
. The “:” in the first set and the “ ” in the second are both
read “such that” and serve the same purpose. The symbols are interchangeable,
and both solution sets are valid.
7.37 Express the solution to the inequality in set notation.
Begin by isolating the absolute value expression left of the inequality symbol.
Express the absolute value inequality as two inequalities that do not include
absolute value expressions, as explained in Problems 7.31 and 7.33.
x ≥ 13 or x ≤ –13
The solution set of the inequality is {x : x ≤ –13 or x ≥ 13}. Like in Problem 7.36,
you can replace the colon in the solution set with a short vertical bar to get the
equally valid solution set .
The solution
set basically says,
“Any real number x
is a solution if that
number x is greater
than –23 and less
than –1.”
Drop the
bars to get one
inequality; then
drop the bars, reverse
the symbol, and take
the opposite of 13 to
get the other.
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