5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book May 8, 2018 13:46

52 STEP 4. Review the Knowledge You Need to Score High

Example 3

Here is a summary of the different types of intervals on a number line:

INTERVAL NOTATION SET NOTATION GRAPH

[a,b] {x|a≤x≤b} a[ b]

(a,b) {x|a<x<b} a( b)

[a,b) {x|a≤x<b} a[ b)

(a,b] {x|a<x≤b} a( b]

[a,∞) {x|x≥a} a[

(a,∞) {x|x>a} a(

(−∞,b] {x|x≤b} b]

(−∞,b) {x|x<b} b)

(−∞,∞) {x|xis a real number}

Solving Absolute Value Inequalities

Letabe a real number such thata≥0.

|x|≥a⇔(x≥aorx≤−a) and|x|>a⇔(x>aorx<a)

|x|≤a⇔(−a≤x≤a) and|x|<a⇔(−a<x<a)

Example 1

Solve the inequality| 3 x− 6 |≤15 and sketch the solution on the real number line.

The given inequality is equivalent to

− 15 ≤ 3 x− 6 ≤ (^15) []
− 9 ≤ 3 x ≤ 21 −^37
− 3 ≤ x≤ 7
Therefore, the solution set is the interval [−3, 7] or, in set notation,{x|− 3 ≤x≤ 7 }.
Example 2
Solve the inequality| 2 x+ 1 |>9 and sketch the solution on the real number line.
The inequality| 2 x+ 1 |>9 implies that
2 x+ 1 >9or2x+ 1 <− 9 4
( )
− 5
Solving the two inequalities in the above line, you havex>4orx<−5. Therefore, the
solution set is the union of the two disjoint intervals (x>4)∪(x<−5) or, by writing the
solution in set notation,{x|(x>4) or (x<−5)}.