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

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

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

Example 1

Solve the inequality

2 x− 5

x− 3

≤1.

1. Rewrite:
   2 x− 5
   x− 3

− 1 ≤ 0

2. Combine:
   2 x− 5 −x+ 3
   x− 3

≤ 0 ⇔

x− 2

x− 3

≤ 0

3. Set the numerator and denominator equal to 0 and solve forx:x=2 and 3.


4. Determine intervals:
   (−∞, 2), (2, 3) and (3,∞)
   2 3


5. Select anx-value and evaluate the fraction:

SELECTED FACTOR FACTOR FRACTION

INTERVAL x-VALUE (x−2) (x−3)

x− 2

x− 3

(−∞,2) 0 −− +

(2, 3) 2.5 +− −

(3,∞)6 ++ +

Therefore, the interval (2, 3) makes the fraction<0.

6. Check the endpoints: Atx=3, the fraction is undefined. Thus, the only endpoint is
   x=2. Since the inequality is less than or equal to 0,x=2 is included in the solution.


7. Write the solution: The solution is the interval [2, 3).

Example 2

Solve the inequality

2 x− 5

x− 3

≤1 by using a graphing calculator. (See Figure 5.2-5.)

ToolsF1 TraceF3

MAIN RAD AUTO FUNC

ZoomF2 ReGraphF4 MathF5 DrawF6 F7Pen

[−7.9, 7.9] by [−3.8, 3.8]

Figure 5.2-5

1. Entery 1 =
   2 x− 5
   x− 3
   andy 2 =1.


2. Find the intersection points:x=2. (Note that atx=3,y 1 is undefined.)


3. Determine the intervals on whichy 1 is belowy 2 : The interval is (2, 3).