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Solve the inequality
\frac{3-z}{z+1} \ge 2(z + 4).
Write your answer in interval notation.

 Jan 18, 2025
 #1
avatar+28 
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3zz+12(z+4)

 

Multiply both sides by z+1: 3z2(z+4)(z+1)

 

When we simplify, we get: 3z2z2+10z+8

 

Take everything to one side: 2z2+11z+50

 

Now, we can factor the quadratic: (2z+1)(z+5)0. Clearly, the critical points are z=1/2,5, and 1 (because that is what is undefined in the original equation)

 

We can now test values below 5, between 5 and 1, between 1 and 1/2 and values greater than 1/2, and see which intervals satisfy the inequality.

 

Substiuting 6 in (interval below 5): 9/54 - True

Substituting 2 in (interval between 5 and 1): 58 - Not True

Substituting 3/4 in (interval between 1 and 1/2): 1513/2 - True

Substituting 0 in (interval greater than 1/2): 38 - Not True

 

There is two intervals which satisfy the original inequality, so our answer In interval notation is (,5)(1,1/2)

 

  

 Jan 19, 2025
 #2
avatar+8 
0

Ahh...Owinner. the sign is greater and EQUAL to, so -5 and -1/2 are actually valid solutions there.

the answer is

(,5](1,12]

 Jan 20, 2025

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