1. Solving Rational Inequalities
Part 1

2. In order to solve a rational inequality,
it MUST be set equal to ZERO and be
simplified to a SINGLE RATIONAL
EXPRESSION.
Example:
𝒙 𝟐 −𝟗 𝒙 𝟐 −𝟏 <𝟎

3. 𝒙 𝟐 −𝟗 𝒙 𝟐 −𝟏 <𝟎
Now follow these steps…
Step 1: Factor the top and bottom of the rational expression and find all the values of x that make the top = 0 and the bottom = 0.
(top) (x + 3) (x – 3) so… x = 3 and -3
(bottom) (x + 1) (x – 1) so… x = 1 and -1

4. Step 2: Draw a picture of the x-axis and mark these
points. These are often called “critical points”.
Discuss marking the critical points with open or closed circles depending on the inequality symbol.

5. Step 3: Our critical points partition the x-axis into
five intervals.
Pick a point (your choice!) within each interval. Let’s use x = 0, x = ±2, and x = ±4.
Compute f(x) for these points.
𝑓 −4 = 𝑓 −2 =
𝑓 0 = 𝑓 2 =
𝑓 4 =
−5 3 <0
7 15 >0
−9 −1 >0
−5 3 <0
7 15 >0

6. These five points represent what happens in the intervals in which they are contained.
You can indicate this on the x-axis by inserting plus or minus signs on the x-axis. + -- + -- +

7. Step 4: We want the expression to be
LESS THAN 0, so we are looking for the
intervals that are negative.
The solution is …. (-3, -1) and (1, 3).
If the inequality had ≥ or ≤ then you would use brackets [ ] around the intervals.

8. x – 1 x + 2 ≤ 0
(-2, 1]
(-∞, -3] , [-2,-1) , (5, ∞)
x 2 + 5x + 6 x 2 − 4x – 5 ≥ 0
x 2 − 16 x 2 − 3x + 2 < 0
(-4,1) , (2,4)

9. Solving Rational Inequalities
Part 2
What do I do when the problem isn’t a SINGLE RATIONAL EXPRESSION set equal to ZERO?

10. Solving Rational Inequalities 2
Remember from previous notes that you must COMBINE all fractions together on ONE SIDE so it is set to 0.
2 𝑥−4 + 1 𝑥+1 >0
How would we combine this into a single fraction on the left?

11. Try this one:
1 𝑥 𝑥−4 <0

12. What if there are terms on BOTH sides?
2𝑥 4 − 5𝑥+1 3 >3

13. Extra Credit Level:

14. Practice: 1) 2) 3) 4) 5)
4 𝑥−6 + 2 𝑥+1 >0
𝑥+6 4𝑥 −3 ≥1
𝑥 2 −𝑥−11 𝑥−2 ≤3
1 𝑥 > 1 𝑥+5
𝑥 𝑥+1 − 𝑥−1 𝑥 < 1 20