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Solve the inequality:

 

\(\frac{3-z}{z+1} \ge 1 \)

 Mar 18, 2020
 #1
avatar+21827 
+1

Multiply both sides by  z+1 to get

 

(BUT there is another consideration.....see Geno's answer below)

 Mar 18, 2020
edited by ElectricPavlov  Mar 18, 2020
edited by ElectricPavlov  Mar 18, 2020
edited by ElectricPavlov  Mar 18, 2020
 #2
avatar+18837 
0

To solve  (3 - z) / (z + 1) >= 1

you need to consider two situations:

1) when z + 1 is positive, and

2) when z + 1 is negative.

For situation 1:  z + 1 > 0   --->   z > -1

Solving by multiplying both sides by z + 1   --->   3 - z  >= z + 1   --->   2 >= 2z   --->   z <= 1

But, we must combine that with the restriction that z > -1 to get:  -1 < z <= 1

Now, for the situation when z + 1 is negative   --->   z + 1 < 0   --->   z < -1

Multiplying both sides by z + 1 (since z + 1 is negative, we must change the direction of the inequality)

--->   3 -z <= z + 1   --->   2 <= 2z   --->   1 <= z   --->   z >= 1   This is impossible, so we must reject this possibility!

 

We have to consider both possibilities because we have to change the direction of the inequality.

 Mar 18, 2020
 #3
avatar+21827 
0

Thanx for the correction , Geno!

ElectricPavlov  Mar 18, 2020

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