Two unknowns in two inequalities

Let’s tackle this question! What don’t we know?

We are given two clues:

Clue 1: y > 1

Clue 2: 3x > y

Since we need to solve for x/y, first we’ll use the clue that has both unknowns (x and y) in it.

We need to take 3x > y and make it into something it currently is not. (Don’t try to do this with people!) What do we want to make it into? x/y, since that’s the original question. Trust me, we can do this!

To get there, we need to divide both sides of 3x > y by y, but let’s pause for a moment. Before you divide or multiply an inequality, ask yourself if you are dividing or multiplying by a negative number. If you are, the inequality will flip. (Are you kidding me? you may be thinking. How will I learn this? To which I heartily reassure you that we’ll cover it in the next blog post!)

We ask ourselves, for this problem, if we divide by y, are we dividing by a negative number? Let’s look at the first clue: y > 1. Oh thank heavens, y is positive. We can continue on.

3x > y

Divide both sides by y:

The left side will stay as it is. On the right side, y divided by y is 1. (Anything divided by itself is 1. Anything fits perfectly inside itself exactly one time.)

We are getting closer to our goal of having x/y on one side. What is ruining the party? That 3!

Since the left side is being multiplied by 3, to get rid of it we need to divide both sides by 3. Remember, inequalities are like twins, if you give one a cookie, you better give the other one a cookie. Since we need to divide the left side by 3, that means we need to divide the right side by 3 as well.

On the left side the 3 divided by 3 is 1. (Anything divided by itself is 1. Anything fits perfectly inside itself exactly one time.)

Well this is progress! Let’s remind ourselves what the question was to begin with.

What is our answer? Of the four options, which option is greater than 1/3?

 That’s right, it is C! 

If you’re not seeing why the answer is C, a number line may help.

Good work!