Solving Rational Inequalities Part 1

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

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

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.

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

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. + -- + -- +

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.

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)

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

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?

Try this one: 1 𝑥+2 + 3 𝑥−4 <0

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

Extra Credit Level:

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