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 integers...less 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. 

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

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

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.

More practice dealing with signs this time with absolute value thrown in the mix.

Last year, a certain company began manufacturing product X and sold every unit of product X that it produced.

Last year, a certain company began manufacturing product X and sold every unit of product X that it produced. Last year the company's total expenses for manufacturing product X were equal to $100,000 plus 5 percent of the company's total revenue from all units of product X sold. If the company made a profit on product X last year, did the company sell more than 21,000 units of product X last year?

(1) The company's total revenue from the sale of product X last year was greater than $110,000.

(2) For each unit of product X sold last year, the company's revenue was $5.

This is a very tough and time consuming question from the GMAT Official Guide (video solution below). It has a few things that make it extra challenging:

  1. It's a long word problem so there's a lot to chew through.
  2. It has multiple variables.
  3. It's an inequality/threshold question

That said, conceptually it's not hard at all. The basic idea: find the break even point or threshold and see if the statements help nail down whether you are above or below.

As always start by reading carefully. Do not rush to the statements. Data Sufficiency success is all about setup!

Let's go ahead and define the question: Did the company sell more than 21,000 units of product X last year?

You might be tempted to stop there. But: keep going! How will we determine whether the company sold more than 21,000 units? What does that 21,000 depend on? What's that you say???? The price? YES! This all hinges on the price per unit. So let's get the inequality defined so we can figure out what price per unit of product X gets us above 21,000 units.

Here's this part translated into an inequality: Last year the company's total expenses for manufacturing product X were equal to $100,000 plus 5 percent of the company's total revenue from all units of product X sold.

#*$ - (100,000 + 5/100*#*$) > 0

Next add the number of units for which you want the dollar amount. We want to know what $ comes out to with # = 21,000.

21,000*$ - 100,000 - 5/100*21,000*$ > 0

21,000*$ - 100,000 -1050*$  > 0

19950*$ -100,000 > 0

$ > 100,000/19950

That comes out to a little more than 5. So at 21,000 units that price is a bit more than $5. So if we're 5 or below we've sold more than 21,000 units.

Let's take a look at statement (1) The company's total revenue from the sale of product X last year was greater than $110,000.

Always write out the algebra: #*$ > 110,000,

We need to know if # > 21,000. With statement (1) we can't separate # from $. So:

Yes: 110,000 units for $1 each.

No: 1 unit for $110,000

Insufficient.

Statement (2) For each unit of product X sold last year, the company's revenue was $5.

This helps. We defined the break even for 21,000 units at 100,000/19950 or a shade above 5. So if we know that revenue per unit X is $5 then the company has sold more than 21,000 units (as price goes down units sold goes up). Sufficient.

B.

Last year, a certain company began manufacturing product X and sold every unit of product X that it produced. Last year the company's total expenses for manufacturing GMAT Explanation Diagram

Last year, a certain company began manufacturing product X Video Solution

Additional GMAT Data Sufficiency Thresholds, Yes/No, Inequality Practice Questions

These two questions are a bit different in content and not as wordy as the above but they share the bigger concept of defining the threshold/breakeven/inequality.

GMAT Data Sufficiency Thresholds/Inequalities Question of the Day 1

GMAT Question of the day Thresholds/Inequalities/Statistics

 

GMAT Question of the Day - Data Sufficiency - Algebra/Threshold

In a certain company 2/5 of the employees are either engineers or scientists. What is the ratio of engineers to scientists?

(1) There are 75 employees in the company

(2) There are more than 3 times as many engineers as there are scientists

AnswerShow

GMAT Question of the Day Solution

It's good policy on GMAT Data Sufficiency to spend as much time as necessary to understand the question before moving on to the statements. The questions (and the information given in the question) are your friends. In this question of the day there isn't a ton of information in the question but still it might help to write out that 2/5's of the company is made up of Engineers and Scientists and that we are looking for E/S.

Statement 1 tells us that that E + S = 30. But we don't know what the mix is. Insufficient.

Statement 2 tells us that the ratio is greater than 3 to 1. So the ratio could be 3:1 or 4:1 or 7:2. Many possibilities. Insufficient.

Putting the statements together provides some limitations but not enough of them to narrow the ratio down. We could have 29 engineers and 1 scientist or 28 engineers and 2 scientists. Both ratios are greater than 3 to 1. Multiple possibilities. Insufficient.

 

GMAT Question of the Day - Data Sufficiency - Inequalities

If m and z are positive numbers is 30m/(m+z) + 50z/(m+z) greater than 39?

(1) z/m > 1

(2) m/(m+z) < 1/2

AnswerShow

GMAT Question of the Day Solution:

Look at this as a weighted average question. If m = z then the answer will end up in the middle of 30 and 50. In this case 40. So the real question here is: z > m?  If this is confusing then I would suggest trying some numbers to make this concept very clear. Make  z=m, z > m, and finally m > z. Hopefully that will shed some light on how this question works (I worked this out at the bottom).

(1) Simplify the algebra to z > m. If z is greater than m then the value of the expression must be closer to 50 than 30. So the expression must be greater than 39. Sufficient.

(2) If the first part of the expression has less than half of the weight than the total expression must be closer to 50 then 30. So the expression must be greater than 39. Sufficient.

GMAT Question of the Day Data Sufficiency Inequalities Diagram

GMAT Question of the Day - Data Sufficiency - Number Properties

If a, b, and c are not equal to zero, is abc > 0?

(1) ab + ac > 1

(2) ac + cb > 1

AnswerShow

Here's a problem solving question also dealing with testing signs: How many of the integers that satisfy the inequality (x+2)(x+3)/(x−2) ≥ 0 are less than 5?

And another couple GMAT Question of the day examples featuring number properties/positive negative/signs

GMAT Question of the Day Number Properties/Signs

GMAT Question of the Day Absolute Value/Number Properties/Signs

GMAT Question of the Day Solution

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