How do you solve and write the following in interval notation:#((x+8)(x-5) )/ (x-1) ≥ 0#?
The solution is
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Therefore,
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To solve and write the inequality (\frac{{(x+8)(x-5)}}{{x-1}} \geq 0) in interval notation, follow these steps:
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Find the critical points by setting the numerator and denominator equal to zero and solving for (x). ((x+8)(x-5) = 0) gives (x = -8) and (x = 5). (x - 1 = 0) gives (x = 1).
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Plot these critical points on a number line.
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Test intervals between the critical points by choosing test points in each interval and determining if the expression is positive or negative.
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Based on the signs of the expression in each interval, determine the intervals where the expression is greater than or equal to zero.
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Write the solution in interval notation.
The critical points divide the number line into four intervals: ((-∞, -8)), ((-8, 1)), ((1, 5)), and ((5, +∞)).
Testing:
- Choose (x = -9) in ((-∞, -8)): (\frac{{(-9+8)(-9-5)}}{{-9-1}} = \frac{{(-1)(-14)}}{{-10}} = \frac{{14}}{{10}} > 0).
- Choose (x = 0) in ((-8, 1)): (\frac{{(0+8)(0-5)}}{{0-1}} = \frac{{(8)(-5)}}{{-1}} = \frac{{-40}}{{-1}} < 0).
- Choose (x = 2) in ((1, 5)): (\frac{{(2+8)(2-5)}}{{2-1}} = \frac{{(10)(-3)}}{{1}} = \frac{{-30}}{{1}} < 0).
- Choose (x = 6) in ((5, +∞)): (\frac{{(6+8)(6-5)}}{{6-1}} = \frac{{(14)(1)}}{{5}} > 0).
The expression is greater than or equal to zero in the intervals ((-∞, -8)) and ((5, +∞)).
Therefore, the solution in interval notation is: ((-∞, -8) \cup (5, +∞)).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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