# How do you solve #(4x-5)/(x+3)>0#?

There are now two permissible cases:

Subtract 12 from both sides to get:

Divide both sides by 4 to get:

Subtract 12 from both sides to get:

Divide both sides by 4 to get:

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To solve the inequality (4x - 5) / (x + 3) > 0:

- Find the critical points by setting the numerator and denominator equal to zero and solving for x.
- Determine the sign of the expression in each interval created by the critical points.
- Identify the intervals where the expression is greater than zero.

The critical points are where the numerator (4x - 5) and the denominator (x + 3) are equal to zero.

Numerator: 4x - 5 = 0 x = 5/4

Denominator: x + 3 = 0 x = -3

Now, we have three intervals: (-∞, -3), (-3, 5/4), and (5/4, ∞).

Test a value in each interval to determine the sign of the expression:

- Choose x = -4 for (-∞, -3)
- Choose x = 0 for (-3, 5/4)
- Choose x = 2 for (5/4, ∞)

For x = -4: (4(-4) - 5) / (-4 + 3) = (-21) / (-1) = 21 > 0, so the interval (-∞, -3) is positive.

For x = 0: (4(0) - 5) / (0 + 3) = (-5) / 3 < 0, so the interval (-3, 5/4) is negative.

For x = 2: (4(2) - 5) / (2 + 3) = (3) / 5 > 0, so the interval (5/4, ∞) is positive.

Therefore, the solution to the inequality is: x ∈ (-∞, -3) U (5/4, ∞).

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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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