In the MGMAT book, it mentions that when multiplying by unknown variables, take both positive/negative scenarios into account
4/x<1/3?
1. 12<x
2. little confused here. do I take x to be -x?
4/-x<1/3
12<-x
-12>x?
?????
multiplying by variables in inequalities
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Hey Gurpinder,
Great question, and one that's pretty important. When using variables and inequalities, you need to consider whether the variable is negative...you don't multiply the variable by -1.
In your example, consider that the following two values of x both satisfy the inequality:
4/x < 1/3
Well, any negative value of x turns that whole term negative, so -1 would give us -4 < 1/3 ---> true
And any value of x that makes 4/x smaller then 4/12 (which is 1/3) also works, so x could be 18: 4/18 < 1/3 because 2/9 is less then 1/3.
___________________________________________
Now, algebraically, what you want to do is recognize that you don't know whether x is positive or negative. If it's positive, when performing the algebra you don't change the sign:
4/x < 1/3 becomes
4 < x/3
12 < x
But if x were negative, you would have to flip the sign. So:
4/x < 1/3 becomes
4 > x/3
12 > x (BUT remember - this is only if x is negative, so 9, 10, or 11 don't work...this really just means that any negative number less than 12 (so all negative numbers) also satisfies the equation).
So...what's important is that you consider that there are negative values that also satisfy the inequality. This happens most often in Data Sufficiency, where often testing a number or two on the negative side helps to bear this out conceptually maybe even more than just using the algebra (which is admittedly a little awkward above).
Great question, and one that's pretty important. When using variables and inequalities, you need to consider whether the variable is negative...you don't multiply the variable by -1.
In your example, consider that the following two values of x both satisfy the inequality:
4/x < 1/3
Well, any negative value of x turns that whole term negative, so -1 would give us -4 < 1/3 ---> true
And any value of x that makes 4/x smaller then 4/12 (which is 1/3) also works, so x could be 18: 4/18 < 1/3 because 2/9 is less then 1/3.
___________________________________________
Now, algebraically, what you want to do is recognize that you don't know whether x is positive or negative. If it's positive, when performing the algebra you don't change the sign:
4/x < 1/3 becomes
4 < x/3
12 < x
But if x were negative, you would have to flip the sign. So:
4/x < 1/3 becomes
4 > x/3
12 > x (BUT remember - this is only if x is negative, so 9, 10, or 11 don't work...this really just means that any negative number less than 12 (so all negative numbers) also satisfies the equation).
So...what's important is that you consider that there are negative values that also satisfy the inequality. This happens most often in Data Sufficiency, where often testing a number or two on the negative side helps to bear this out conceptually maybe even more than just using the algebra (which is admittedly a little awkward above).
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.
GMAT Instructor
Chief Academic Officer
Veritas Prep
Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.