Inequalities SUMMARY EXERCISES on Solving Linear and Absolute Value Equations and
The equation and inequalities just described are examples of absolute value
equations and inequalities.They involve the absolute value of a variable expression
and generally take the form
or
where kis a positive number. From FIGURES 29–31, we see that
has the same solution set as or
has the same solution set as or
has the same solution set as and
Thus, we solve an absolute value equation or inequality by solving the appropriate
compound equation or inequality.
|x| 64 x7- 4 x 6 4.
|x| 74 x6- 4 x 7 4,
|x| = 4 x=- 4 x= 4,
|axb|k, |axb|>k, |axb|<k,
Solving Absolute Value Equations and Inequalities
Letkbe a positive real number and pandqbe real numbers.
Case 1 To solve solve the following compound equation.
or
The solution set is usually of the form which includes two
numbers.
Case 2 To solve solve the following compound inequality.
The solution set is of the form which is a disjoint
interval.
Case 3 To solve solve the following three-part inequality.
The solution set is of the form 1 p,q 2 ,a single interval.
k<axb<k
|axb|<k,
1 - q,p 2 ́ 1 q,q 2 ,
axb>k or axb<k
|axb|>k,
5 p,q 6 ,
axbk axbk
|axb|k,
p q
p q
pq
NOTE Some people prefer to write the compound statements in Cases 1 and 2 of the
preceding box as follows.
or Alternative for Case 1
and or Alternative for Case 2
These forms produce the same results.
OBJECTIVE 2 Solve equations of the form for.
Remember that because absolute value refers to distance from the origin, an ab-
solute value equation will have two parts.
|axb|k, k> 0
ax+ b 7 k - 1 ax+ b 27 k
ax+ b= k - 1 ax +b 2 = k
