# How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5? GMAT Explanation

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

A. 1
B. 2
C. 3
D. 4
E. 5

If you start doing the algebra, working the inequality/quadratic, this can get ugly. On GMAT quant remember to stay practical. Not everything has a tidy algebraic solution. Some things do so let’s not completely forget about solving equations/inequalities but, again, let’s just make good decisions and use the tools that make sense for the job. In this case we have a single variable inequality and a somewhat limiting constraint for the value of x: How many of the integersless than 5.

My gut would be to start testing numbers less than 5: 4, 3, 2, 1, 0, -1, -2, -3…

The inequality we’re attempting to satisfy, (x+2)(x+3)/(x−2) ≥ 0, hinges on the expression being positive or negative. With that in mind I’d pay special attention to signs. We don’t really care about the actual value of the expression just whether we are above or below zero. Context is key!

Here’s a video if you need to brush up on multiplying positive and negative numbers. If though you are missing that fundamental you might want to take a step back in your GMAT prep and dig back into the basics. This is a pretty advanced question to tackle if you’re having trouble with signs.

OK – back to work! Popping in anything 3 or greater is positive so 4 and 3 are good. 2 yields a zero in the denominator so that’s not going to work because diving by zero is undefined. -1 yields negative. -2 and -3  each yield zero so both work. -4 yields negative. So does -5 and everything smaller than that. So -2, -3, 3, and 4 all satisfy the inequality. D. ## Additional Algebra, Inequality, Positive/Negative, Picking Numbers GMAT Practice Questions

This GMAT Question of the DS Number Properties/Signs example is a bit different in the specifics especially because it’s Data Sufficiency but the focus on positive/negative is spot on.

Here’s the another Data Sufficiency example question with testing signs. Again, a little different but very similar in certain important aspects.