Question 30039: x-2/x+6 >= 3
Please help me solve this equation
Answer by Fermat(136)   (Show Source):
You can put this solution on YOUR website! Multiply everything by x
giving,
x² - 2 + 6x >= 3x (assuming x is positive)
rearrange,
x² + 3x - 2 >= 0
use the quadratic formula,
a = 1, b = 3, c = -2
x = 0.5616, x = -3.5616
(x-0.5616)(x+3.5616) >= 0
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Since the product of the two terms is positive (>=0), then the terms must be both positive or both negative (negative times negative = positive).
So,
x >= 0.5616 and x >= -3.5616
i.e. x >= 0.5616
================
or
x =< 0.5616 and x =< -3.5616
i.e. x =< -3.5616
================
But x was assumed to be positive (see third line). So the last solution (x =< -3.5616) is invalid since it requires x to be negative)
In fact, if you assume x to be negative, you get,
x² - 2 + 6x <= 3x (notice the change in the inequality sign)
rearrange,
x² + 3x - 2 <= 0
giving,
(x-0.5616)(x+3.5616) <= 0
==========================
This requires one term to be positive and the other term to be negative - at the same time.
So,
x =< 0.5616 and x >= -3.5616
or
-3.5616 =< x =< 0.5616
but x requires to be negative, so this becomes,
-3.5616 =< x =< 0
=================
The two solutions are,
x >= 0.5616
===========
and
-3.5616 =< x =< 0
=================
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