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:
- \(x - 10 = 0 \Rightarrow x = 10\)
- \(x - 17 = 0 \Rightarrow x = 17\)
- \(x + 10 = 0 \Rightarrow x = -10\)
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:
- \((- \infty, -10)\)
- \((-10, 10)\)
- \((10, 17)\)
- \((17, \infty)\)
We will test a point in each interval:
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For the interval \((- \infty, -10)\) (let's test \(x = -11\)): \[ (-11 - 10)(-11 - 17)(-11 + 10) = (-21)(-28)(-1) = -588 \quad (\text{negative}) \]
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For the interval \((-10, 10)\) (let's test \(x = 0\)): \[ (0 - 10)(0 - 17)(0 + 10) = (-10)(-17)(10) = 1700 \quad (\text{positive}) \]
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For the interval \((10, 17)\) (let's test \(x = 15\)): \[ (15 - 10)(15 - 17)(15 + 10) = (5)(-2)(25) = -250 \quad (\text{negative}) \]
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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":
- The function is 0 at \(x = -10\), \(x = 10\), and \(x = 17\).
- 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 \([-10, 10] \cup [17, \infty)\).
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