Inequalities: Choosing the correct value
This type of GRE problem asks you to choose which value the unknown could be worth.
Steps to Success
What is the unknown? (It’s usually a variable like x or y.)
Get the unknown to appear only once.
Now get the unknown alone:
a. Handle any numbers that are being added or subtracted to the unknown.
b. Then handle any numbers that are being multiplied by or divided by the unknown.
Draw a number line. (See post about number lines if you need a refresher!)
Check your work
Looking at the inequality, let’s run through the steps:
7 < 3x – 2 < 16
Step 1: What is the unknown? It is x.
Step 2: Get the unknown to appear only once.
Do you see “x” more than once here: 7 < 3x – 2 < 16? Nope! That step is already done.
Step 3: Now get the unknown alone. Right now the 3 and the 2 are spoiling things for us.
First we handle any numbers that are being adding or subtracted to x. What is being adding or subtracted? The 2! Since it’s being subtracted, we are going to add it everywhere. (Why everywhere? Think of this inequality like triplets. You can only give one of them a balloon if you’re going to give EACH of them their own balloon.
7 < 3x – 2 < 16
+2 +2 +2
9 < 3x < 18
Great! So now we have 9 < 3x < 18. That 3 is spoiling things for us! Since x is being multiplied by 3, we’ll divide by 3 everywhere.
3 < x < 6
This is looking good! We got x alone.
Step 4: Draw a number line.
Let’s revisit the original question, putting in our simplified inequality:
If 3 < x < 6, which of the following could be the value of x?
a) 2
b) 3
c) 4
d) 6
e) 8
Look at our number line. What option is shaded in green? Clearly not 2 or 8, so we can cross out option a and e.
We are left with:
b) 3
c) 4
d) 6
It may be tempting to think it could be 3, or 6, but those would only be true if it was 3 < x < 6 with the “or equals to” option with the inequality symbols. Since it is 3 < x < 6, that means x is less than 6 (not less than or equal to 6), and x is greater than 3 (not greater than or equal to 3). That is why we have the donut holes over 3 and 6 in the number line above, to show those numbers are not solutions. We can cross out options b and d.
c) 4
We are left with c) 4, which is in the shaded green part of our number line.
Let’s check our work. It takes a few seconds, and will build your confidence while taking the GREs. Take the original inequality (not the one we’ve simplified, just in case we made an error along the way!):
7 < 3x – 2 < 16
Plug our answer of 4 in for x, and see if it holds.
7 < 3(4) – 2 < 16
7 < 12 – 2 < 16
7 < 10 < 16; yes this a true statement. We got it right!
Alternate ending:
What if you had picked an incorrect answer, how would checking your work help? Let’s say you thought it was option b) 3
Take the original inequality:
7 < 3x – 2 < 16
Plug our answer of 3 in for x, and see if it holds.
7 < 3(3) – 2 < 16
7 < 9 – 2 < 16
7 < 7 < 16; this does not hold. 7 isn’t greater than 7.
Now we would know we made a mistake. If this happens to you, just remember there is no need to panic. You are solving GRE math problems, not performing open-heart surgery - no one dies because of a mistake on a GRE problem! Just take a fresh sheet of paper, copy the problem onto it, and start again. The more you practice, the faster you will be, which will give you more time to check your work and find any errors.
Have a great day!