A rational expression
looks like a fraction
and has a variable
in the denominator. A rational inequality
just means we now have a rational expression
combined with a
or sign. Below are some examples of what a rational inequality
might look like.
Solving rational inequalities requires the same initial step as solving quadratic equations; we MUST get all terms on the left side
of the inequality
sign and have zero on the right side
of the inequality
sign. Once all terms are on the left side
of the inequality, we have to make sure we only have a single rational expression.
By having a rational expression
compared to zero (with a
sign), we will only need to find where the expression
is equal to zero (set the top = 0) or where the expression
is undefined (set the bottom = 0). Since the expression
can change signs (from positive to negative or from negative to positive) at these points we can call them changing points.
When we place the changing points on a number line, we will obtain intervals along the number line. We will substitute a test value from each interval into the inequality
to see if the inequality
is true or false. If it is true, then all values in the interval will make the inequality
true. If it is false, then all values in the interval will make the inequality
Summary of steps:
- Write the inequality so that there is a single rational expression on the left side of the inequality and zero on the right side of the inequality.
- Determine the changing points by setting the numerator equal to zero and setting the denominator equal to zero.
- Use the changing points to separate the number line into intervals.
- Select test values in each interval and substitute those values into the inequality.
- If the test value makes the inequality true, then the entire interval is a solution to the inequality.
- If the test value makes the inequality false, then the entire interval is not a solution to the inequality.
- Express your answer in interval notation.