How do we solve quadratic inequalities? Do Now: What is the difference between an equation and an inequality?

What are we trying to find when we solve inequalities? We are trying to find the values of x that make the inequality true. If we are trying to solve the inequality x2-x-6<0, then we want the values of x that would make y negative in the equation y=x2-x-6 We can look at the graph and see that this is true when -2<x<3

But how can we solve it algebraically? First we want to factor. (x-3)(x+2)<0 Next we solve for values of x that make the expression equal zero x=3, x=-2 Now we put this on a number line

x2-x-6<0 -2 3 Since the inequality is simply less than zero, then we use an open circle to mark the points. Note: If the inequality was less than or equal to zero, the the circle would be solid Now, we pick a value in each section; x<-2, -2<x<3, 3<x and plug it into the inequality.

x2-x-6<0 - (-3-3)(-3+2)=(-6)(-1)=6 (0-3)(0+2)=(-3)(2)=-6 -2 3 - + + (-3-3)(-3+2)=(-6)(-1)=6 (0-3)(0+2)=(-3)(2)=-6 (4-3)(4+2)=(1)(6)=6 Finally, we express our answer in one of two ways: Symbolically: -2<x<3 Graphically: Shade in the correct part of the number line

Example Given: x2-2x-20>4 Find the solution set. Graph the solution set on a number line.

Summary/HW If the problem is a quadratic, is there always a set pattern of positives and negatives? HW pg 101, 1-16 even