# Solving an inequality and multiplying by -1

I need to solve the following inequality and I have a couple of questions:

$\frac {1}{3} (x+5)>2x+10$

I get the process up untill $-x>5$ (assuming that's correct, please correct me if I'm wrong, lol!). I now need to multiply both sides by -1 to get a positive x.

According to a friend of mine, multiplying by -1 with inequalities causes the inequality sign (or whatever it's called) to switch. Meaning I'll get $x<-5$.

To me that just doesn't make any logical sense because I'm doing the exact same on each side of the ">"-sign, meaning I am not changing the relation between the two numbers? So why would the "greater than" become "less than"?

Is he correct? Is the answer $x<-5$? If so, what's the logic behind this?

• Please help me get the 1/3 fraction correct, lol. – Julian Nikolay Krogh-Fredrikse Oct 14 '15 at 0:32
• Try \frac{1}{3} – Henricus V. Oct 14 '15 at 0:33
• $\dots -3 < -2 < -1 < 0 < 1 < 2 < 3 \dots$ – BrianO Oct 14 '15 at 0:37

He is correct. By multiplying $-1$ to both sides, you are mirroring the two numbers across $0$.(On the real line) Numbers on the "left" side of a number becomes the "right" side, and vise versa.