Question

Use algebraic​ and/or graphical methods to solve the inequality.
​(xminus14​)(xminus17​)(xplus14​)greater than or equals0
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ solution(s) is/are xequals
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
B.
The solutions are the values of x in the interval
  
enter your response here.
​(Type your answer in interval​ notation.)
C.
There is no solution.

Answer 1

To solve the inequality \((x - 14)(x - 17)(x + 14) \geq 0\), we first find the critical points by setting each factor equal to zero.

1. \(x - 14 = 0 \) gives \(x = 14\)
2. \(x - 17 = 0\) gives \(x = 17\)
3. \(x + 14 = 0\) gives \(x = -14\)

Thus, the critical points are \(x = -14\), \(x = 14\), and \(x = 17\).

Next, we'll determine the sign of the expression \((x - 14)(x - 17)(x + 14)\) in the intervals defined by these critical points:
- \((- \infty, -14)\)
- \((-14, 14)\)
- \((14, 17)\)
- \((17, \infty)\)

### Test the intervals:

1. **Interval \((- \infty, -14)\)**: Choose \(x = -15\).
\[
(-15 - 14)(-15 - 17)(-15 + 14) = (-29)(-32)(-1) < 0
\]

2. **Interval \((-14, 14)\)**: Choose \(x = 0\).
\[
(0 - 14)(0 - 17)(0 + 14) = (-14)(-17)(14) > 0
\]

3. **Interval \((14, 17)\)**: Choose \(x = 15\).
\[
(15 - 14)(15 - 17)(15 + 14) = (1)(-2)(29) < 0
\]

4. **Interval \((17, \infty)\)**: Choose \(x = 18\).
\[
(18 - 14)(18 - 17)(18 + 14) = (4)(1)(32) > 0
\]

### Summary of intervals:
- In \((- \infty, -14)\), the expression is negative.
- In \((-14, 14)\), the expression is positive.
- In \((14, 17)\), the expression is negative.
- In \((17, \infty)\), the expression is positive.

### Include the critical points:
At the critical points \(x = -14\), \(x = 14\), and \(x = 17\), the expression equals zero:
- \((x - 14)(x - 17)(x + 14) = 0\)

### Solution:
From our tests, the expression is non-negative in the intervals:
- \([-14, 14]\)
- \([17, \infty)\)

Combined, the solution in interval notation is:
\[
[-14, 14] \cup [17, \infty)
\]

### Final answer:
B. The solutions are the values of \(x\) in the interval \([-14, 14] \cup [17, \infty)\).