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

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

Review of Precalculus 53

Example 3

Solve the inequality| 1 − 2 x|≤7 and sketch the solution on the real number line.

The inequality| 1 − 2 x|≤7 implies that

− 7 ≤ 1 − 2 x≤ 7
− 8 ≤ − 2 x≤ 6
4

[]
4 ≥ x≥− 3 −^3
− 3 ≤ x≤ 4

Therefore, the solution set is the interval [−3, 4] or, by writing the solution in set notation,
{x|− 3 ≤x≤ 4 }. (See Figure 5.2-2.)

ToolsF1 TraceF3

MAIN RAD AUTO FUNC

ZoomF2 ReGraphF4 MathF5 DrawF6 F7Pen

[−7.9, 7.9] by [−5, 10]

Figure 5.2-2

Note that you can solve an absolute value inequality by using a graphing calculator. For
instance, in Example 3, entery 1 =| 1 − 2 x|andy 2 =7. The graphs intersect atx=−3 and 4,
andy 1 is belowy 2 on the interval (−3, 4). Since the inequality is≤, the solution set is [−3, 4].

Solving Polynomial Inequalities

1. Write the given inequality in standard form with the polynomial on the left and zero on
   the right.


2. Factor the polynomial, if possible.


3. Find all zeros of the polynomials.


4. Using the zeros on a number line, determine the test intervals.


5. Select anx-value from each interval and substitute it in the polynomial.


6. Check theendpointsof each interval with the inequality. Use a parenthesis if the endpoint
   is not included and a bracket if it is.


7. Write the solution to the inequality.

Example 1

Solve the inequalityx^2 − 3 x≥4.

1. Write in standard form:x^2 − 3 x− 4 ≥ 0


2. Factor the polynomial: (x−4)(x+1)


3. Find zeros: (x−4)(x+1)=0 implies thatx=4 andx=−1.


4. Determine intervals:

(−∞,−1) and (−1, 4) and (4,∞)

− (^14)