Solving Inequalities in Interval Notation: Where Did I Go Wrong?

[aeroengphys]

In interval notation, I was asked to solve for 5 < 2x - 1 ≤ 15

I did this by making the 4 cases...

2x-1>5 or 2x-1<-5 AND 2x-1≤ 15 and 2x-1≥-15
x > 3 or x < -2 AND x ≤ 8 and x ≥ -7

So for my final answer...i got [-7,-2),(3,8]

But when i check it, none of the numbers between -7 and -2 check...what'd I do wrong?

Lastly, is the answer actually just (3,8]?

 

[neutrino]

What's the use in checking if 2x-1<-5 and 2x-1>= -15?

 

[aeroengphys]

honestly...I have no idea, but we were taught to make four cases for those...so that's how I've been doing it ever since.

 

[aeroengphys]

Actually, I just realized that I could've solved for x by adding 1 to the left and right, and dividing by 2 to the left and right...which gives me an answer of 3 < x ≤ 8...i'm pretty sure that's right...guess i went about it the wrong way

 

[neutrino]

Actually, I just realized that I could've solved for x by adding 1 to the left and right, and dividing by 2 to the left and right...which gives me an answer of 3 < x ≤ 8

Yup, that's probably the easiest way.

 

[James R]

I have no idea, but we were taught to make four cases for those...so that's how I've been doing it ever since.

You're probably confusing it with a problem like:

$$2 < |2x - 1| \le 15$$

where the absolute value sign creates a problem.