Question

How do I solve this inequality?

Answer 1

Okay, so starting out, I'm just gonna type the inequality.

-2 (5 + 6n) < 6 (8 - 2n)

Alright! So to solve any inequality, you typically want to get the varaible by itself on one side. In this case, our varaible is "n". So how do we get it alone? Well, it always helps to simplify both sides as much as you can. We have some multiplication we can do!

I'm not sure if you're familiar with this, but if I have

-2 (5 + 6n)

I can rewrite this (using the distribution property) as:

-2×5 + -2×6n

I can apply that same logic to the other side. Rewriting this, I would have:

-2×5 + -2×6n < 6×8 + 6×-2n

Alright! At this point, we can further simplify both sides by doing any kind of combining. I see a lot of multiplication, so let's go ahead and do that part. I'm just gonna jump to doing all the multiplication, but if you're unsure on what I've done, just let me know in a comment.

anyway! now we have:

-10 - 12n < 48 -12n

Now, we need to move stuff on their proper sides. We want all "n" values on one side, and all others on the other. I'm gonna add 10 on both sides to move it to the right.

-12n < 48+10 -12n

Next, I'm going to add 12n on both sides so that value is now on the left.

-12n + 12n < 48 +10

Now, for combining like terms! 48 and 10 can go together to form 58.

-12n + 12n <58

Now what about 12n and -12n? Those values added together results in 0n, or just 0.

0 <58

So...now what? Well, our n's cancelled, so either what n equals doesn't matter and the inequality is allows false, or its always true. To find out, we see what the inequality is actually saying. In this, it's saying 0 is less than 58. That IS true, right? So that means, for all values of n, our inequality is true!

Hope that helps, and if any of it doesn't make sense, don't hesitate to ask!