SOLUTION: If xy > 1 and z < 0, which of the following statements must be true? I. x > z II. xyz < -1 III. xy/z < 1/z

Algebra ->  Inequalities -> SOLUTION: If xy > 1 and z < 0, which of the following statements must be true? I. x > z II. xyz < -1 III. xy/z < 1/z       Log On


   

Question 308695: If xy > 1 and z < 0, which of the following statements must be true?

I. x > z
II. xyz < -1
III. xy/z < 1/z
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I. False. For example, let x = -10 and z = -1. Clearly x%3Ez is false. If we let y=-5, then xy=-10%28-5%29=50%3E1 showing that xy%3E1 is true.


II. False, this is only true if xyz%3Cz. So xyz+%3C+-1 is on the right track, but it is false. For example, if xy=2, and we let z=-1%2F2, then xyz=2%28-1%2F2%29=-1 which is clearly not less than -1. We must make the requirement that the right side be 'z' and not -1.

III. This is true since dividing both sides of an inequality by a negative number will flip the inequality sign. Basically, divide both sides of xy+%3E+1 by the negative number 'z' to get %28xy%29%2Fz+%3C+1%2Fz (don't forget to flip the sign).