Solving Rational Inequalities

Rational

A Rational Expression looks like:

Rational Expression

Inequalities

Sometimes we need to solve rational inequalities like these:

Symbol
Words
Example
>
greater than
(x+1)/(3-x) > 2
<
less than
x/(x+7) < -3
greater than or equal to
(x-1)/(5-x) ≥ 0
less than or equal to
(3-2x)/(x-1) ≤ 2

Solving

Solving inequalities is very like solving equations ... you do most of the same things.

Graph of Rational Inequality
When we solve inequalities
we try to find interval(s),
such as the one I have marked "<0"

These are the steps:

  • find "points of interest":
    • the "=0" points (roots), and
    • "vertical asymptotes" (where the function is undefined)
  • in between the "points of interest", the function is either greater than zero (>0) or less than zero (<0)
  • then pick a test value to find out which it is (>0 or <0)

Here is an example:

Example: (3x-10)/(x-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-10)/(x-4) - 2 > 0

Then multiply 2 by (x-4)/(x-4):

(3x-10)/(x-4) - 2(x-4)/(x-4) > 0

Now we have a common denominator, let's bring it all together:

(3x-10-2x+8)/(x-4) > 0

Simplify:

(x-2)/(x-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 plot!

But here is the plot of (x-2)/(x-4) so you can see:

Graph of Inequality