Algebra book Chapter 11 Question 3

[kyen09]

Question 3.
If 4/x < 1/3, what is the possible range of values for x?

I was able to figure out that x>12. However, when I look at the solution, it showed that I should have consider a 2nd case as well. The 2nd case was to consider the possibilities of x being negative. Thus the 2nd solution would be 12> x.

I understand how the 2nd solution is reach. But I don't understand why I need to consider the possibility of X being negative. My understanding of any questions having a 2nd (Negative) solution is when X is square (or having any even numbers). But in this case, the X was only 1. So I assume there would only be 1 answer.

Thanks in advance.

[RonPurewal]

Question 3.
If 4/x < 1/3, what is the possible range of values for x?

I was able to figure out that x>12. However, when I look at the solution, it showed that I should have consider a 2nd case as well. The 2nd case was to consider the possibilities of x being negative. Thus the 2nd solution would be 12> x.

I understand how the 2nd solution is reach. But I don't understand why I need to consider the possibility of X being negative. My understanding of any questions having a 2nd (Negative) solution is when X is square (or having any even numbers). But in this case, the X was only 1. So I assume there would only be 1 answer.

Thanks in advance.

Hi,
You posted this in the wrong folder on the forum, so we moved it for you.

The best answer to "I don't understand why I need to consider the possibility of X being negative" is... well, plug in some negative numbers and watch what they do!
that's the best way to actually learn this stuff: the "hands-on" way, in which you just start throwing numbers into things and watching what they do. once you've seen enough examples of how different numbers behave in different situations, you'll have a better intuition about this kind of stuff.
(if you think you need to have a "rule" memorized for every possible situation, you are not going to have much fun with this exam.)

the more general point here is that you should essentially always consider signs when dealing with inequalities.
yes, there are some inequalities on which signs don't really matter... but there are a lot more of them (especially on the gmat exam) on which signs do matter, a lot. so, you may as well just think about signs every time.

[kyen09]

Correct me if I'm wrong in my thinking.

What I got out of this question is that the variable x is a denominator for a fraction and since I am moving a variable to another side of the inequality, I should consider the possibility that X could be either positive/negative. Thus this is why it has to be done.

Also, I noticed in chapter 6 inequalities (pg92 middle bold) stats that you cannot multiply or divide an inequality by a variable, unless you know the sign of the number that the variable stands for. Am I reading this statement incorrectly? or should I assume that because we did 2 possibilities for the solution (Positive/Negative), thus we were able to get around it.

[jnelson0612]

Correct me if I'm wrong in my thinking.

What I got out of this question is that the variable x is a denominator for a fraction and since I am moving a variable to another side of the inequality, I should consider the possibility that X could be either positive/negative. Thus this is why it has to be done.

Also, I noticed in chapter 6 inequalities (pg92 middle bold) stats that you cannot multiply or divide an inequality by a variable, unless you know the sign of the number that the variable stands for. Am I reading this statement incorrectly? or should I assume that because we did 2 possibilities for the solution (Positive/Negative), thus we were able to get around it.

Yes. Anytime you multiply or divide an inequality by a variable you must consider the possibility that the variable could be negative or positive, unless you have a constraint eliminating one of these possibilities.

In this case:
1) x is positive:
4/x < 1/3
Cross multiply:
12 < x

2) x cannot be zero, because that we create an undefined number. So let's proceed.

3) x is negative
4/x < 1/3
If I cross multiply here, I want to flip my sign, because I am multiplying by a negative.
12 > x

Huh. Does that really make sense? If x is 8, the statement is not true. Actually no positive numbers under 12 work. 0 doesn't work. Keep going. Oh, wait! All negative values of x will work because they will turn the left side into a negative number.

So my ultimate answer is that x is either greater than 12 or less than 0.