Let f ( x ) = 7 + | 2 x - 1 | ....

Misael Matthews

Misael Matthews

Answered

2022-06-25

Let f ( x ) = 7 + | 2 x - 1 | . How do you find all x for which f ( x ) < 16 ?

Answer & Explanation

Xzavier Shelton

Xzavier Shelton

Expert

2022-06-26Added 26 answers

Step 1
Given: f ( x ) = 7 + | 2 x - 1 | and f ( x ) < 16
We can write the inequality:
7 + | 2 x - 1 | < 16
Subtract 7 from both sides:
| 2 x - 1 | < 9
Because of the piecewise definition of the absolute value function,
| A | = { A ; A 0 - A ; A < 0
we can separate the inequality into two inequalities:
- ( 2 x - 1 ) < 9 and 2 x - 1 < 9
Multiply both sides of the first inequality by -1:
2 x - 1 > - 9 and 2 x - 1 < 9
Add 1 to both sides of both inequalities:
2 x > - 8 and 2 x < 10
Divide both sides of both inequalities by 2:
x > - 4 and x < 5
This can be written as:
- 4 < x < 5
To check, I will verify that the end points equal 16:
7 + 2 ( - 4 ) - 1 ) | = 7 + | - 9 = 16
7 + | 2 ( 5 ) - 1 | = 7 + | 9 | = 16

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