College Algebra 9th txtbk

SECTION 1–3 Absolute Value in Equations and Inequalities 67
EXAMPLE 2 Solving Absolute Value Problems Geometrically
Interpret geometrically, solve, and graph. For inequalities, write solutions in both inequality
and interval notation.
(A) x 3 5 (B) x 3 6 5
(C) 0 6 x 3 6 5 (D) x 3 7 5
SOLUTIONS
(A) The expression |x 3| represents the distance between x and 3, so the solutions to
|x 3| 5 are all numbers that are exactly 5 units away from 3 on a number line.
x 3 5 2 or 8
The solution set is {2, 8}.
This is not interval notation.
2
5 5
3 8
x
(B) Solutions to |x 3| 5 are all numbers whose distance from 3 is less than 5. These
are the numbers between 2 and 8:
2 6 x 6 8
The solution set is (2, 8).
This is interval notation.
( )
2 3 8
(C) Expressions like 0 |x 3| 5 are important in calculus. The solutions are all
numbers whose distance from 3 is less than 5, and is not zero. This excludes 3 itself
from the solution set:
x
2 6 x 6 8
x 3
or
(2, 3) (3, 8)
Hole
( )
2 3
8
x
(D) The solutions to x 3 7 5 are all numbers whose distance from 3 is greater than
5; that is,
x 6 2
or
x 7 8
(, 2) (8, )
)
(
2 3 8
ZZZ CAUTION ZZZ
The pair of inequalities 2 x and x 8 can be written as a double inequality:
2 6 x 6 8 or in interval notation (2, 8)
But the pair x 2 or x 8 from Example 2(D) cannot be written as a double
inequality, or as a single interval: no number is both less than 2 and greater
than 8.