Question

Use algebraic​ and/or graphical methods to solve the inequality.
​(xminus10​)(xminus17​)(xplus10​)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 \((x - 10)(x - 17)(x + 10) \geq 0\), let's first identify the critical points where the expression equals zero. The critical points occur when each factor is equal to zero:

1. \(x - 10 = 0 \Rightarrow x = 10\)
2. \(x - 17 = 0 \Rightarrow x = 17\)
3. \(x + 10 = 0 \Rightarrow x = -10\)

Thus, the critical points are \(x = -10\), \(x = 10\), and \(x = 17\).

Next, we need to determine the signs of the product \((x - 10)(x - 17)(x + 10)\) in each of the intervals defined by these critical points. The intervals to test are:

1. \((- \infty, -10)\)
2. \((-10, 10)\)
3. \((10, 17)\)
4. \((17, \infty)\)

We will test a point in each interval:

1. **For the interval \((- \infty, -10)\)** (let's test \(x = -11\)):
\[
(-11 - 10)(-11 - 17)(-11 + 10) = (-21)(-28)(-1) = -588 \quad (\text{negative})
\]

2. **For the interval \((-10, 10)\)** (let's test \(x = 0\)):
\[
(0 - 10)(0 - 17)(0 + 10) = (-10)(-17)(10) = 1700 \quad (\text{positive})
\]

3. **For the interval \((10, 17)\)** (let's test \(x = 15\)):
\[
(15 - 10)(15 - 17)(15 + 10) = (5)(-2)(25) = -250 \quad (\text{negative})
\]

4. **For the interval \((17, \infty)\)** (let's test \(x = 18\)):
\[
(18 - 10)(18 - 17)(18 + 10) = (8)(1)(28) = 224 \quad (\text{positive})
\]

Now we summarize the results:

- \((- \infty, -10)\) : negative
- \((-10, 10)\) : positive
- \((10, 17)\) : negative
- \((17, \infty)\) : positive

Next, we include the critical points in our solution since the inequality is "greater than or equal to zero":

1. The function is 0 at \(x = -10\), \(x = 10\), and \(x = 17\).
2. The intervals where the function is greater than or equal to zero are:
- \([-10, 10]\) (which is positive)
- \([17, \infty)\) (which is positive)

Thus, the overall solution in interval notation is:
\[
[-10, 10] \cup [17, \infty)
\]

Based on your options, the correct choice is:
**B. The solutions are the values of \(x\) in the interval** \([-10, 10] \cup [17, \infty)\).