# How do you find the critical points for the inequality #(2x+1)/(x-9)>=0#?

See the explanation.

I think the question wants the key numbers or the partition numbers.

graph{y=(2x+1)/(x-9) [-25.9, 39.05, -23.36, 9.1]}

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To find the critical points for the inequality (\frac{{2x+1}}{{x-9}} \geq 0), we first need to determine where the expression is equal to zero and where it is undefined.

Setting the numerator equal to zero: [2x + 1 = 0] [2x = -1] [x = -\frac{1}{2}]

Setting the denominator equal to zero: [x - 9 = 0] [x = 9]

Now, we have two critical points: (x = -\frac{1}{2}) and (x = 9). We need to test the intervals between and beyond these critical points to determine where the expression is positive or negative.

Testing the interval (x < -\frac{1}{2}): Choose a test point (x = -1). Substitute it into the inequality: [\frac{2(-1)+1}{(-1)-9} = \frac{-1}{-10} > 0] Since the expression is positive in this interval, it satisfies the inequality.

Testing the interval (-\frac{1}{2} < x < 9): Choose a test point (x = 0). Substitute it into the inequality: [\frac{2(0)+1}{(0)-9} = \frac{1}{-9} < 0] Since the expression is negative in this interval, it does not satisfy the inequality.

Testing the interval (x > 9): Choose a test point (x = 10). Substitute it into the inequality: [\frac{2(10)+1}{(10)-9} = \frac{21}{1} > 0] Since the expression is positive in this interval, it satisfies the inequality.

Therefore, the critical points where the inequality holds true are (x \leq -\frac{1}{2}) and (x \geq 9).

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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.

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