math help?

Question

3z +5 < -7/5z+4

solve the inequality for z. Simplify as much as possible.

Answer 1

It depends on which question you're asking. I don't know which of the following three possibilities you mean:

A. 3z + 5 < -(7/5) z + 4
<=> 3z + (7/5) z < 4 - 5
<=> (22/5) z < -1
<=> z < -5/22.

B. 3z + 5 < -7 / (5z) + 4
<=> 3z + 7 / (5z) < 4 - 5
<=> 15z^2 + 7 < -5z and z > 0, OR 15z^2 + 7 > -5z and z < 0
<=> 15z^2 + 5z + 7 < 0 and z > 0, OR 15z^2 + 5z + 7 > 0 and z < 0; note that if z > 0 then 15z^2 + 5z + 7 > 7 > 0, so the first case is impossible
<=> 15z^2 + 5z + 7 > 0 and z < 0
Now the discriminant of 15z^2 + 5z + 7 is 5^2 - 4(15)(7) = -395 < 0, so there are no zeros for this quuadratic; since the leading term has a positive coefficient it must be positive everywhere. In other words 15z^2 + 5z + 7 > 0 is true for all z.
<=> z ∈ R and z < 0
<=> z < 0.

B. 3z + 5 < -7 / (5z + 4)
<=> (3z + 5) (5z + 4) < -7 and 5z + 4 > 0, OR (3z + 5) (5z + 4) > -7 and 5z + 4 < 0
<=> 15z^2 + 37z + 27 < 0 and z > -4/5, OR 15z^2 + 37z + 27 > 0 and z < -4/5
The discriminant of 15z^2 + 37z + 27 is -251, so this quadratic is also positive everywhere.
<=> z < -4/5.

Answer 2

<=> 3z+7/5z<4-5
<=>22/5z<-1
<=>z<-5/22

Answer 3

3z + 5 < -7/5z + 4
3z + 7/5z < 4 - 5
22/5z < -1
z < -5/22

Answer 4

3z+7/5z<-5+4
22/5z<-1
22<-5z
22/5<-z
-22/5