0

# Quarter Wit, Quarter Wisdom: When Can You Divide by a Variable?

We have often come across test takers confused about division by a variable. When is it allowed, when is it not allowed? Why is it allowed in some cases and not in others? What are the constraints we need to look out for? For example: Is division by x allowed here: x^2 = 10x?
Is division by x allowed here: y = 4x?
Is division by x allowed here: x^2 < 4x?
Let’s take a detailed look at all these questions today. The basic guidelines: Let’s look at the three questions given above and try to solve them using these guidelines: Is division by x allowed here: x^2 = 10x? The first thing to find out here is whether or not x can equal 0. Case 1: If no other information has been given, then x can be 0 and we cannot divide by it. This is how we proceed in that case: x^2 – 10x = 0
x(x – 10) = 0
x = 0 or 10
Case 2:  If the question stem tells us that x is not 0, then we can divide by x. x^2/x = 10x/x
x = 10
Obviously, we don’t get the second solution (x = 0) in this case, as we already know that x cannot be 0. Now let’s look at the second problem: Is division by x allowed here: y = 4x? Again, this is an equation and we need to know whether or not x can equal 0. Case 1: If x can be 0, you cannot divide by it. In this case, x = 0 and y = 0 is one of the infinite possible solutions. Case 2: If the question stem states that x cannot be 0, then we can do the following: y/x = 4 Now let’s look at the final question: Is division by x allowed here: x^2 > -4x? Here, we have an inequality. Before deciding whether we can divide by x or not, we need to know not only whether x can be equal to 0, but also whether x is positive or negative. Case 1: If we know nothing about the possible values that x can take, then this is how we proceed: x^2 + 4x > 0
x(x + 4) > 0
Now we can use the method discussed in the first problem to arrive at the range of x. x > 0 or x < -4 Case 2: If we know that x is positive, then we can proceed like this: x^2/x > -4x/x
x > -4
Since we are given that x is positive, we know that that x > 0 (looking at the two options above). Case 3: If we know that x is negative, then this is how we will proceed: x^2/x < -4x/x (we flip the sign of the inequality because we divide by x, which is negative)
x < -4
The results obtained are logical, right? When x can be anywhere on the number line, we get the range as x > 0 or x < -4. If x has to be positive, the range is x > 0.
If x has to be negative, the range is x < -4.
Getting ready to take the GMAT? We have   running all the time. And, be sure to follow us on  , , and  ! , a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the   for Veritas Prep and regularly participates in content development projects such as  !
This article first appeared here Quarter Wit, Quarter Wisdom: When Can You Divide by a Variable?