SECTION 2.7 Absolute Value Equations and Inequalities 117
Solving Special Cases of Absolute Value EquationsSolve each equation.
(a)
See Case 1 in the preceding box. The absolute value of an expression can never
be negative,so there are no solutions for this equation. The solution set is
(b)
See Case 2 in the preceding box. The expression will equal 0 onlyif
Add 3.Divide by 7.The solution of this equation is Thus, the solution set is^37. E^37 F,with just one element.
x=
3
7
.
7 x= 3
7 x- 3 = 0
| 7 x- 3 |
| 7 x- 3 |= 0
0.
| 5 x- 3 |=- 4
EXAMPLE 7
Check by
substituting in the
original equation.NOW TRYSolving Special Cases of Absolute Value InequalitiesSolve each inequality.
(a)
The absolute value of a number is always greater than or equal to 0.Thus,
is true for allreal numbers. The solution set is
(b)
Add 3 to each side.There is no number whose absolute value is less than so this inequality has no so-
lution. The solution set is
(c)
Subtract 4 from each side.The value of will never be less than 0. However, will equal 0 when
x=7.Therefore, the solution set is 576.
|x- 7 | |x- 7 |
|x- 7 |... 0
|x- 7 |+ 4 ... 4
0.
- 2,
|x+ 6 |6- 2
|x+ 6 |- 3 6- 5
|x| Ú- 4 1 - q, q 2.
|x| Ú- 4
NOW TRY EXAMPLE 8
EXERCISE 8
Solve each inequality.
(a)
(b)
(c) |x- 2 |- 3 ...- 3
| 4 x+ 1 |+ 564|x|7- 10NOW TRY
EXERCISE 7
Solve each equation.
(a)
(b)| 7 x+ 12 |= 0
| 3 x- 8 |=- 2Absolute value is used to find the relative errorof a measurement. If represents the
expected measurement and xrepresents the actual measurement, then the relative error
in xequals the absolute value of the difference between and x, divided by
In quality control situations, the relative error often must be less than some predeter-
mined amount. For example, suppose a machine filling quartmilk cartons is set for a
relative error no greater than0.05. Here the relative error and
we must find x, given the following condition.
No greater thantranslates as.For Discussion or Writing
With this tolerance level, how many ouncesmay a carton contain?
` ...
32 - x
32
` ...0.05
xt=32 oz, = 0.05 oz,
relative error in x= `
xt-x
xt
`
xt xt.
xt
CONNECTIONS
NOW TRY ANSWERS
- (a) (b)
- (a) 1 - q, q 2 (b) 0 (c) 526
0 E-^127 FNOW TRY