1. ## Solving Inequality

Can someone help me simplify the following compound inequality and express the result using interval notation:
7<3(x+2) +10<25 AND 8>10-2x>0

2. ## Inequality

7<3(x+2) +10<25 AND 8>10-2x>0

(7 < 3x + 6 + 10 < 25) and (8 > 10 - 2x > 0)
7 < 3x + 16 < 25 and -2 > -2x > -10
-9 < 3x < 9 and -1 < x < 5
-3 < x < 3 and nothing new here, so:

It is really your choice of what particluar interval notation you prefer.
As far as I can tell over 20 years of algebra texts, different authors
have different preferences. Not having a lot of time to make fancy
text here with, let's go with this:

The solution set is the set of all x such that
x is betwenn -3 and 3
AND
x is between -1 and 5.

Do these two intervals overlap? If so, that would be where x is statisfies the inequality, if such a set exists. In my so-called mind (actually, some part of my brain) I see a number line:

-3 -2 -1 0 1 2 3 4 5

On inspecting the number line above in regard to the two constraints I conclude that the two intervals do overlap:

Lower bound for x approaches -1 from the right.

Upper bound for x approaches 3 from the left.

So solution set = {x:-1<x<3}

I hope I didn't make a stupid mistake along the way.

I could possibly find out by substituting approriate values for x into the orignal inqualities above and check if:

(members of the solution set work for both inequalities at the same time)

AND

(values outside of the solution set did not work for both inequalities at the same time).

But that takes more time than I have now.

It's 11:15 PM here and I have to get up at 5:30 AM to go to work.