x+2/(x+4)(x-4) <0?

Answer 1

Oops, singingrose forgot to consider the different cases.

I'm assuming you mean:

(x + 2)/[(x + 4)(x - 4)] < 0

This basically represents a number divided by 2 other numbers. The 3 factors divide the number line into 4 regions:

Case 1: x < - 4

In this case, all 3 numbers would be negative. A negative number divided by the product of two negative numbers will result in a negative number (< 0), so any x < -4 works.

Case 2: -4 < x ≤ -2

In this region, (x + 2) and (x - 4) are negative while (x + 4) is positive. Therefore the quotient will be positive and there are no solutions in this region.

Case 3: -2 ≤ x < 4

In this region, only (x - 4) is negative, so the quotient is negative. Any number in this range works.

Case 4: x > 4

In this region, all three factors are positive, so the quotient is positive. No numbers in this range work.

Therefore, the actual answer is:

x < -4 or -2 ≤ x < 4

Or, in interval notation,

x ∈ (-∞, -4) U [-2, 4)

Answer 2

(x + 2) /[(x + 4)(x - 4)] < 0

Our critical values are what make the left hand side EQUAL to zero. They are: x = {-2, -4, 4}

Now, we determine the behaviour around these values. We're interested in these four intervals:
a) (-infinity, -4)
b) (-4, -2)
c) (-2, 4)
d) (4, infinity)

We can test ANY one value in each interval. Since our inequality is less than 0, we want a negative result after testing.

a) Test x = -10. {since -10 lies in (-infinity, -4). Then,
(x + 2) /[(x + 4)(x - 4)] = (negative) / [(negative)(negative)], which is equal to a negative number. Keep the interval (-infinity, -4).

b) Test x = -3. Then
(x + 2) /[(x + 4)(x - 4)] = (-1) / [(1) (-7)] = 1/7, which is a positive number. Reject this interval.

c) Test x = 0. Then
(x + 2) /[(x + 4)(x - 4)] = (0 + 2) / [(4)(-4)] = -2/4, which is negative. Keep the interval (-2, 4).

d) Test x = 100. Then
(x + 2) /[(x + 4)(x - 4)] = (positive)/[positive*positive], which is positive. Reject this interval.

Therefore, our solution set is

(-infinity, -4) U (-2, 4)

Answer 3

First: eliminate the fraction - multiply the denominator by everything...

(x+4)(x-4)[x+2/(x+4)(x-4)] < (x+4)(x-4)(0)

Sec: cross cancel "like" terms & combine the remaining terms...

x + 2 < 0

Third: isolate "x" on one side-subtract "2" by both sides (when you move a variable to the opposite side, always use the opposite sign - subtraction goes with addition)...

x + 2 - 2 < 0 - 2

x < - 2