+0

# Solve the inequality 3-z/z+1≥1

0
52
2

Solve the inequality 3-z/z+1≥1

Apr 27, 2020

#1
+1

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

Apr 27, 2020
#2
+21017
+1

(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.

Apr 27, 2020