Question 616808
{{{(x-1)/(x+2)>0}}}
The solution to this is based on some simple logic:<ul><li>This inequality, in its essence, says that a fraction is positive.</li><li>There are only two situations that make a fraction positive:<ul><li>The numerator and denominator are both positive; or</li><li>The numerator and denominator are both negative.</li></ul></li></ul>All we have to do is translate this logic into some inequalities and solve.<br>
"the numerator and denominator are both positive" translates into:
((x-1) > 0 and (x+2) > 0)
"the numerator and denominator are both negative" translates into:
((x-1) < 0 and (x+2) < 0)
"the numerator and denominator are both positive or the numerator and denominator are both negative" translates into:
((x-1) > 0 and (x+2) > 0) or ((x-1) < 0 and (x+2) < 0)<br>
Now we solve.
(x > 1 and x > -2) or (x < 1 and x < -2)<br>
(x > 1 and x > -2) will simplify. The only way x can be greater than 1 and -2 at the same time is if x > 1. So we can replace (x > 1 and x > -2) with just (x > 1).<br>
Using similar logic on (x < 1 and x < -2) we find that we can replace it with just (x < -2)<br>
So our "reduced" solution is:
(x > 1) or (x < -2)<br>