3-z/z+1≥1 Multiply both sides by z+1
3-z >= z+1 add z to both sides
3 >= 2z + 1 subtract one from both sides
2 >= 2z divide by 2
z<=1
+23245 (3 - z) / (z + 1) >= 1
I want to multiply both sides by (z + 1) but I have to be careful because I'll have to change the direction of the inequality sign
if I multiply by a negative. So I get two cases.
Case 1: (z + 1) > 0 ---> [ (3 - z) / (z + 1) ] · (z + 1) >= 1 · (z + 1)
z > -1 3 - z >= z + 1
2 >= 2z
1 >= z ---> z <= 1
Final answer for this case: -1 < z and z <= 1 or -1 < z <=1 or (-1, 1]
Case 2: (z + 1) < 0 ---> [ (3 - z) / (z + 1) ] · (z + 1) <= 1 · (z + 1) (switch the inequality sign)
z < -1 3 - z <= z + 1
2 <= 2z
1 <= z ---> z >= 1
This is impossible because z can't be both < -1 and >= 1 at the same time.
The final answer is what we got in case 1.
