Calculus- Early Transcendentals, 2021a

14 Review
Looking where the + symbols are located gives the solution:
(
−∞, −9 − √ ) (
33 ⋃
−2, −9 + √ )
33
6
6
When writing the final answer we use exact expressions for numbers in mathematics, not approximations
(unless stated otherwise).
♣
1.1.5 The Absolute Value
The absolute value of a number x is written as |x| and represents the distance x is from zero. Mathematically,
we define it as follows:
{ x, ifx ≥ 0
|x| =
−x, ifx < 0
Thus, if x is a negative real number, then −x is a positive real number. The absolute value does not just
turn minuses into pluses. That is, |2x − 1| ≠ 2x + 1. You should be familiar with the following properties.
Absolute Value Properties
1. |x|≥0.
2. |xy| = |x||y|.
3. |1/x| = 1/|x| when x ≠ 0.
4. |−x| = |x|.
5. |x + y|≤|x| + |y|. This is called the triangle inequality.
6. √ x 2 = |x|.
Example 1.15: √ x 2 = |x|
Observe that √ (−3) 2 gives an answer of 3, not −3.
When solving inequalities with absolute values, the following are helpful.
Case 1: a > 0.
• |x| = a has solutions x = ±a.
• |x|≤a means x ≥−a and x ≤ a (that is, −a ≤ x ≤ a).