SECTION 2.7 Absolute Value Equations and Inequalities 115
Check that the solution set is The graph consists of the single interval shown
in FIGURE 34.
1 - 4, 3 2.
–5 –4 –3 –2 –1 0 1 2 3 4
FIGURE 34CAUTION When solving absolute value equations and inequalities of the types
in Examples 1, 2, and 3,remember the following.
1. The methods described apply when the constant is alone on one side of the
equation or inequality and is positive.
2. Absolute value equations and absolute value inequalities of the form
translate into “or” compound statements.
3. Absolute value inequalities of the form translate into “and”
compound statements, which may be written as three-part inequalities.
4. An “or” statement cannotbe written in three parts. It would be incorrect to
write in Example 2,because this would imply that
which is false.
OBJECTIVE 4 Solve absolute value equations that involve rewriting.
Solving an Absolute Value Equation That Requires RewritingSolve
First isolate the absolute value expression on one side of the equals symbol.
Subtract 5.
Combine like terms.Now use the method shown in Example 1to solve
Subtract 3.Check these solutions by substituting each one in the original equation.
CHECK
Let Let✓ True ✓ True
The check confirms that the solution set is 5 - 10, 4 6. NOW TRY
12 = 12 12 = 12
| 7 |+ 5 12 |- 7 |+ 5 12
| 4 + 3 |+ 5 12 x=4. |- 10 + 3 |+ 5 12 x=-10.
|x+ 3 | + 5 = 12
x= 4 or x=- 10
x+ 3 = 7 or x+ 3 =- 7
|x+ 3 |= 7.
|x+ 3 |= 7
|x+ 3 |+ 5 - 5 = 12 - 5
|x+ 3 |+ 5 = 12
|x+ 3 |+ 5 =12.
EXAMPLE 4
- 772 x+ 177 - 77 7,
|ax+ b| 6 k
|ax+ b| 7 k
NOW TRY
EXERCISE 4
Solve| 10 x- 2 |- 2 =12.
NOW TRYLook back at FIGURES 32, 33 ,AND 34, with the graphs of
and
respectively. If we find the union of the three sets, we get the set of all real numbers.
This is because, for any value of x, will satisfy one and only one of the fol-
lowing: It is equal to 7, greater than 7, or less than 7.
| 2 x+ 1 |
| 2 x+ 1 |=7, | 2 x+ 1 | 7 7, | 2 x+ 1 | 6 7,
NOW TRY
EXERCISE 3
Solve| 4 x- 1 | 6 11.
NOW TRY ANSWERS
3.
- E-^65 ,^85 F
A-^52 , 3B