Solve the inequality
\frac{3-z}{z+1} \ge 2(z + 4).
Write your answer in interval notation.
Multiply both sides by z+1: 3−z≥2(z+4)(z+1)
When we simplify, we get: 3−z≥2z2+10z+8
Take everything to one side: 2z2+11z+5≤0
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): −9/5≥−4 - True Substituting −2 in (interval between −5 and −1): −5≥8 - Not True Substituting −3/4 in (interval between −1 and −1/2): 15≥13/2 - True Substituting 0 in (interval greater than −1/2): 3≥8 - Not True
There is two intervals which satisfy the original inequality, so our answer In interval notation is (−∞,−5)∪(−1,−1/2)
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]