1. Aims: To practice sketching graphs of rational functions To practice sketching graphs of rational functions To be able to solve inequalities by sketching and To be able to solve inequalities by sketching and observation. observation. To be able to solve inequalities by using an algebraic To be able to solve inequalities by using an algebraic method method Graphs Lesson 4

2. Starter Example 2 Solve (x - 2)(3x – 1) ≤ 0 Solve x 2 + x – 3 > 4 x + 1.

3. 3 Remember: When both sides of an inequality are multiplied or divided by a negative number the inequality is reversed. Remember: When both sides of an inequality are multiplied or divided by a negative number the inequality is reversed. –3 < 5 Multiply both sides by –1: 3 < –5 So we have to reverse the inequality sign: 3 > –5 Rules of Inequalities This means that if we were trying to solve the inequality We could not just multiply both sides by as we do not know if this is negative. What we do have to do, is multiply it by, which has to be positive.

4. 4 Methods to Solve Inequalities There are two methods we will look at to solve rational function inequalities. Method 1 Algebraic method: multiply both sides by the denominator squared. Then solve the quadratic inequality. Method 2 Sketching method: sketch the rational function and the line y = 2, then look to see where the curve is greater than the line y = 2. Points of intersection will need to be found.

5. 5 Example 1 Solve the inequality Method 1

6. 6 Method 2 1. Find the intercepts with the axes 2. Find the vertical asymptotes 3. Examine the behaviour as x tends to   Example 1 Solve the inequality x    y 

7. 7 4. Draw line y = 2 see where they cross when

8. Example 2 Solve the inequality Method 1

9. 9 Example 2 Solve the inequality Method 2 1. Find the intercepts with the axes 2. Find the vertical asymptotes 3. Examine the behaviour as x tends to   First sketch x    y 

10. 10 4. Draw line y = 2 see where they cross when

11. 11 1. Solve the inequality (you chose your favourite method or show me both?) On w/b Do ex 5G page 71. then revision ex 5

12. 12