Hello,

I am so stuck...I even watched two hours of videos, but it didn't really help me:(
I am working on these kinds of problems...
3x-4|+5>32
And...as part of our homework, we have to enter it on the computer in interval notation. So I know that I solve it for both sides right? Then I graph it and then I set it in interval notation when I enter my work, this is where I am running into problems. Could you please send me all your work, so I can see what I am doing wrong w/the interval notation part. Should I have two sets of notations for both solutions or should it all be in one. This is my second time taking this class, and it is crucial that I get this down.
Thanks So Much,
Tina

Hi Tina! I can definitely help you with that.

To solve the inequality 3x - 4| + 5 > 32 we need to consider two cases:

Case 1: When the absolute value, | |, is positive:
In this case, we can remove the absolute value sign and solve the inequality as usual. So we have:
3x - 4 + 5 > 32
Simplifying:
3x + 1 > 32
Subtracting 1 from both sides:
3x > 31
Dividing by 3:
x > 31/3

Case 2: When the absolute value, | |, is negative:
In this case, we need to change the sign of the inequality when we remove the absolute value sign. So we have:
-(3x - 4) + 5 > 32
Simplifying:
-3x + 4 + 5 > 32
-3x + 9 > 32
Subtracting 9 from both sides:
-3x > 23
Dividing by -3 and flipping the inequality sign:
x < 23/(-3)
x < -23/3

Now, let's graph these solutions on a number line:

( )
-----------------------------|------------------|----------------
-23/3 31/3

The open circle represents the solutions x < -23/3 and x > 31/3.

Finally, let's write the solution in interval notation:
The interval notation for x < -23/3 is (-∞, -23/3)
The interval notation for x > 31/3 is (31/3, ∞)

Therefore, the overall solution in interval notation is the union of these two intervals:
(-∞, -23/3) U (31/3, ∞)

I hope this helps! If you have any more questions, feel free to ask.