Guide 3 (Equations, Inequalities, VICS) pg. 187 #12

[vinnieregina]

Hello,

I have a question about #12 in Advanced problem set on pg. 187.

The question reads;

If x does not equal zero, is ((x^2+1)/x)>y?

1)x=y
2)y>0

I said statement 1 alone is sufficient, however the manual says the answer is C.

My reasoning is as follows;

If you multiply both sides of the equation by x, you get either of the following two equations:

If x>0, then x^2+1>x^2
If x<0, then x^2+1<x^2

Based on this information I eliminated x<0 as a possibility because any number +1 will always be larger than itself. That being said I determined x>0 and since x=y, Statement 2 appears to simply restate the information in statement 1. Can some please explain where I am going wrong?

Thanks,
Vinnie

[Ben Ku]

If you multiply both sides of the equation by x, you get either of the following two equations:

If x>0, then x^2+1>x^2
If x<0, then x^2+1<x^2

Based on this information I eliminated x<0 as a possibility because any number +1 will always be larger than itself. That being said I determined x>0 and since x=y

The problem is an issue of logic. You eliminated the options where x < 0 because the inequality doesn't make sense. However, rather than eliminating it, this statement really is telling you that if x < 0, then your original inequality is wrong, so the answer is No.

In other words, you really should be saying "If x < 0, then the answer to the question is NO." Likewise, when considering the first possibility, we would say "If x > 0, then the answer to the question is YES" because mathematically it works.

Because we don't know if x is positive or negative, then statement (1) is insufficient.

Hope that makes sense.

[tomslawsky]

I thought of it this way:

since X=Y AND X is positive, you have :

is ((X^2) +1)/X >X

Since X is positive, multiply both sides by X and the inequality (due to X being positive) DOES NOT FLIP.

therefore, X squared plus one in this case is greater than X squared, therefore the answer is "C".

The key is to recognize that since X is positive, you can multiply both sides and retain the direction of the inequality.

At least, I think.

[Ben Ku]

I thought of it this way:

since X=Y AND X is positive, you have :

is ((X^2) +1)/X >X

Since X is positive, multiply both sides by X and the inequality (due to X being positive) DOES NOT FLIP.

therefore, X squared plus one in this case is greater than X squared, therefore the answer is "C".

The key is to recognize that since X is positive, you can multiply both sides and retain the direction of the inequality.

At least, I think.

If you're evaluating both statements together, then we know x is positive because y is positive and x = y. This approach is fine, however, we need to be careful we don't jump to (C) before evaluating (1) and (2) independently.