One unknown in two inequalities

Sometimes life hands you a problem with a nice clean solution. Thankfully this is one of those times! How can you say that? You may be thinking. This is two math problems in one question. My worst nightmare!

Think of problems like these as bookends. One inequality gives you one end, the other inequality gives you the other end, and in between them is the solution. Let’s give it a try!

Where did this p come from? You may be wondering. We always work with x or y! Have no fear, any letter can function as the unknown. The letter is just a symbol. Keep working with p for this problem, and you’ll get a good feel for that.

Let’s take the first inequality:

What don’t we know? We don’t know p. We need to get p alone on one side. Since p is being multiplied by 5, we will divide both sides by 5.

Great, we now know that p > 24. That’s one bookend. Let’s do the other one.

We need to get p alone on one side. Since p is being multiplied by 3, we will divide both sides by 3.

(Bonus tip: you may see 78 and wonder if it’s divisible by 3. To determine if a number is divisible by 3, add up its digits: in this case for 78, we would add 7 + 8. That equals 15, which is divisible by 3, so we know 78 is divisible by 3. Bust this trick out at the next party you go to!)

Okay, now we have our other bookend. We know p < 26 and p > 24.

 You could draw a quick number line to help visualize it:

The problem states that p is an integer, which means that it is a whole number. Looking at our number line, the only whole number it can be is 25. (It can’t be 24 because p > 24, not p > 24. It can’t be 26 because p < 26, not p < 26. Review the post on number lines under Tools You Can Use if you need a quick refresher on that.)

So we’ve determined p is equal to 25.

Looking at our choices:

a.       23

b.      25

c.       26

d.      40

e.       41

Our answer is b, p is equal to 25. Great job!

Checking our work:

Let’s check our work by substituting 25 in for p.

Yes, that holds. 125 is greater than 120. Let’s try the other inequality.

That also holds. 75 is less than 78. Our work checks out!

Why check our work?

IF we had made a mistake and gotten the wrong answer, we would have caught it when we checked our work. Let’s say we made an error and thought the answer was 24. We decide to spend a minute checking our work. We plus 24 in for p in one of the inequalities:

So far so good…let’s try the other inequality.

That does not hold true. 120 is not greater than 120. If you find your work doesn’t check out, take a deep breath, take a fresh piece of paper, and tackle the problem again.

Finding a mistake is already a success; by finding it, you can do something about it.

If you are wondering why this work didn’t check out when one inequality held and one didn’t, just think of it like dating. If two people may go on a date, and one says yes and one says no, there is no date. If there are two inequalities and one holds and one doesn’t, you don’t have a correct answer. (And that’s okay - now you know to go back and try again!)