help in this og 13th edition prob,ineqality

quantskillsgmat wrote:How many of the integers that satisfy the inequality (x+2)(x+3)/(x-2)greater than or equal to 0 are less than 5.
a)1 b)2 c)3 d)4 e)5

The given expression changes its sign at x = -2, x = -3, and x = 2.
Now, for values of x which are less than -3, the expression is always negative as all three terms of the expression are individually negative.

Let us check the sign of the expression for integer values of x which are greater than or equal to -3 but less than 5.

For x = -3, (x + 2)*(x + 3)/(x - 2) = 0 ---------------> YES
For x = -2, (x + 2)*(x + 3)/(x - 2) = 0 ---------------> YES
For x = -1, (x + 2)*(x + 3)/(x - 2) = -2/3
For x = 0, (x + 2)*(x + 3)/(x - 2) = -3
For x = 1, (x + 2)*(x + 3)/(x - 2) = -12
For x = 2, (x + 2)*(x + 3)/(x - 2) = Undefined
For x = 3, (x + 2)*(x + 3)/(x - 2) = 30 ---------------> YES
For x = 4, (x + 2)*(x + 3)/(x - 2) = 21 ---------------> YES

Hence, 4 such integers.

The correct answer is D.

Algebraic Approach:
(x + 2)*(x + 3)/(x - 2) ≥ 0

The expression changes its sign at x = -2, x = -3, and x = 2.
These three values of x divides the number line in four regions. We have to check the sign of the expression for

each of this four region.

x < -3

• All three of (x + 2), (x + 3), and (x - 2) are negative.
  Hence, the expression is negative.

-3 ≤ x ≤ -2

• (x + 3) ≥ 0, (x + 2) ≤ 0 and (x - 2) < 0
  Hence, the expression is greater than or equal to zero.

-2 < x < 2

• (x + 3) > 0, (x + 2) > 0 and (x - 2) < 0
  Hence, the expression is negative.

x > 2

• (x + 3) > 0, (x + 2) > 0 and (x - 2) > 0
  Hence, the expression is positive.

Therefore, the expression is greater than or equal to zero for -3 ≤ x ≤ -2 or x > 2.

Hence, the integer values of x that are less than 5 are -3, -2, 3, and 4.

The correct answer is D.