Source: Official Guide for the GMAT 13th Ed. Problem Solving; #229 Official Guide for the GMAT 2015 14th Ed. Problem Solving; #229

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How many of the integers that satisfy

How many of the integers that satisfy the inequality (x+2)(x+3)/x-2 0 are less than 5?

2 Explanations

5

TANUJ CHAWLA

you can solve the inequality first by factorization and you shall get x>=-2 and x<=-3.
Plot these numbers on number line for x<5(as given in question).
You can extend your list like -4,-3,-2,-1,0,1,2,3,4.
Start putting these values of x in equation given in question and you shall see that the only values of x allowed by the equation are -3,-2-,3,4 so total 4 values and hence the answer

Jan 23, 2017 • Comment

8

Neerja Prashar

Put values of x from -4 to 4
-4 -3 -2 -1 0 1 2 3 4
X ok ok X X X X ok ok

Only four values satisfy the equation

Jun 23, 2015 • Comment

Jonathan , Magoosh Tutor

Right. We know x cannot be less than -3, because then (x+2)(x+3)/(x-2) will be less than 0.

So we only consider the integers between -3 and 4 inclusive.

David Buchanan

So we don't approach this like we would a quadratic inequality problem?

Cydney Seigerman, Magoosh Tutor

While plugging in numbers does work, we could also find the roots and solve the problem that way:

The roots are: x= -3, -2, and 2

Imagining a number line, we can now use these roots to write the ranges of values of x:

-3>x
-3 ? x ? -2
-2 < x < 2
x > 2

Notice that in the third range, x<2 not x?2. x cannot equal 2 because then the original inequality would be undefined.

In a given range, the original expression will be either positive or negative (which we will determine below). Since the problem asks for the values of x for which the expression is greater than zero, the possible solutions will lie in ranges for which the expression is positive. To check the ranges, we have to pick a number that is within that range.

First, let's check -3>x by choosing x= -4:

(-4+2)(-4+3)/(-4-2) = (-2)(-1)/(-6) --> NEGATIVE

Since we end up with a product of three negative numbers, the product will be negative. We do not need to carry out the multiplication to figure this out.

From here, we could test the next range with a value between -3 and -2. Alternatively, we could use the general observation that the sign will alternative as we move from one range to the subsequent range. We can see that by plugging in -2.5:

(-2.5+2)(-2.5+3)/(-2.5-2) = (-.5)(.5)/(-4.5) --> POSITIVE

Again, we don't need to carry through the multiplication to observe that the product will be positive. And since the product is positive, integer values of x that are in the range -3 ? x ? -2 satisfy the conditions of the problem. Therefore, -3 and -2 are possible values of x.

Following this pattern, we can conclude that values of x in the fourth range (x>2) will also satisfy the original inequality. Since x must be less than 5, we must consider integers in the range 2