x=7/5,5
Lets split the absolute value operators:
1) abs(2x+7)
2)abs(6-3x)
There are 4 possibilities:
- 1) can be positive when 2) is positive.
- 1) can be positive when 2) is negative.
- 1) can be negative when 2 is positive.
- 1) can be negative when 2) is negative.
We have to check all these possibilities to find the solution set.
Lets start checking:
2x+7>=0
6-3x>=0
Both should be satisfied for our first condition.
When we solve these inequalities:
A: 2>=x>=(-7/5)
We will need to remember that.
Since we assume both absolute values are positive:
2x+7-(6-3x)=8
5x=7
x=7/5
We assumed that both absolute values are positive. For this to happen, x must have a value in [-7/5,2]. (Look at A:)
7/5 is in the specified range. So it is a member of our solution set.
Lets continue to our work. Our second possibility is: 1) is positive when 2) is negative.
So lets find the range of x
2x+7>=0
6-3x<0
B: x>2
2x+7-(-1)*(6-3x)=8
2x+7+6-3x=8
-x=-5
x=5
x is in the range B: (2,+oo). So it is in our solution set.
Be patient, there are 2 possibilities left.
When 1) is negative, 2) is positive (we assume).
So:
2x+7<0
6-3x>=0
2>=x
x<(-7/5)
As you can see x cannot be greater than 2 while it is less than (-1,4). This means 1) and 2) cannot be negative and positive respectively. There is no value of x to satisfy this condition.
Our final condition: 1) and 2) are both negative.
2x+7<0
6-3x<0
D: 2 < x < (-7/5)
(-1) * (2x + 7) - (-1) * (6 - 3x) = 8
-2x - 7 + 6 - 3x = 8
-5x=9
x=(-9/5)=-1.8
x is not in the range D: (-1.4,2). So 1) and 2) cannot be both negative. There is no value of x to satisfy this condition.
So the solution set is:
SS = {7/5, 5 }