X/x-1≥2 Rational inequality
Answers
Step-by-step explanation:
Example: 3x−10x−4 > 2
First, let us simplify!
But You Cannot Multiply By (x−4)
Because "x−4" could be positive or negative ... we don't know if we should change the direction of the inequality or not. This is all explained on Solving Inequalities.
Instead, bring "2" to the left:
3x−10x−4 − 2 > 0
Then multiply 2 by (x−4)/(x−4):
3x−10x−4 − 2 x−4x−4 > 0
Now we have a common denominator, let's bring it all together:
3x−10 − 2(x−4)x−4 > 0
Simplify:
x−2x−4 > 0
Second, let us find "points of interest".
At x=2 we have: (0)/(x−4) > 0, which is a "=0" point, or root
At x=4 we have: (x−2)/(0) > 0, which is undefined
Third, do test points to see what it does in between:
At x=0:
x−2 = −2, which is negative
x−4 = −4, which is also negative
So (x−2)/(x−4) must be positive
We can do the same for x=3 and x=5, and end up with these results:
x=0 x=2 x=3 x=4 x=5
x−2 < 0 x−2 > 0 x−2 > 0
x−4 < 0 x−4 < 0 x−4 > 0
(x−2)/(x−4) is > 0 0 < 0 undefined > 0
That gives us a complete picture!
And where is it > 0 ?
Less than 2
More than 4
So our result is:
(−∞, 2) U (4, +∞)
We did all that without drawing a
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