1. Inequalities

2. Objectives Understand the laws for working with inequality symbols Solve linear inequalities Solve quadratic inequalities

3. Linear inequalities You can add or subtract a number on both sides of an inequality You can multiply or divide an inequality by a positive number You can multiply or divide an inequality by a negative number, but you must change the direction of the inequality

4. Quadratic Inequalities usual formfactor formcompleted square form Easier form to use with inequalities Intersects the x axis at 2 and 4 Coefficients of x 2 is positive  the parabola bents upwards y is less than 0 2<x<4 interval 7< x <3 Be careful, this does not make sense; x can’t be both (AND) greater than 7 and less than 3. Ok for OR condition Please note that 7 is not less than 3 Please note that 2 is less than 4 An interval (AND) example: Method 1 X must be both greater than 2 AND less than 4

5. Method 2 (tabular method) x<2x=22<x<4x=4x>4 x-2-0+++ x-4---0+ (x-2)(x-4)+0-0+ AND X > 2 AND X < 4

6. For what values of input the output will be less than 0? The graph shows the output (y) of the function for various inputs (x) The graph is below x axis for input values between -2 and 3. in other words the output of the function is negative for input values between -2 and 3 x<-2x=-2-2<x<3x=3x>3 x+2-0++ x-3--0+ (x+2)(x-3)+0-0+ AND X > -2 AND X < 3 Range (interval) bound For what values of input the output will be greater than 0? The graph shows the output (y) of the function for various inputs (x) The graph is above x axis if input values are 3. in other words the output of the function is positive for input values 3 x<-2x=-2-2<x<3x=3x>3 x+2-0++ x-3--0+ (x+2)(x-3)+0-0+ OR X 3 Not range (interval) bound OR X 3 X < -2X > 3 Either condition would satisfy OR instead of AND

7. If two products are positive, then either they both should be positive or they both should be negative Scenario 1Scenario 2 This means x > 3This means x < -2 This is a OR condition (not range/interval bound) If two products are negative, then one should be negative and the other should be positive Scenario 1Scenario 2 This means -2 3This means x < -2 x must be both greater than 3 and less than -2. This AND condition is impossible.

8. Coefficients of x 2 is negative  the parabola bents downwards (vertex at the top) Intersects the x axis at -1 and 5 Y>0 Y<0 Another Example: OR criteria (not range/interval bound) Critical values Either condition would satisfy OR instead of AND

9. Critical values are x= –a and x =+a x<-3x=-3-3<x<3x=3x>3 x-3---0+ x+3-0+++ (x-3)(x+3)+0-0+ Ex: x = -4Ex: x = 2Ex: x = 4 It is easier to assign simple values to a Both are equivalent

10. If two products are negative, one of them MUST be negative and the other MUST be positive Scenario 1Scenario 2 x must be both less than 3 and greater than -1/2. x must be both greater than 3 and less than -1/2. This AND condition is impossible. Another Approach

11. If two products are positive, then either they both should be positive or they both should be negative Scenario 1Scenario 2 This means x > 3 This means x < -1/2 This is a OR condition (not range/interval bound)

12. Don’t have a calculator and can’t factorize the equation or your mean teacher asked you to solve algebraically: complete the square Square means a positive value Equals 0 when x = 2 ; the smallest value There is no value of x for which this condition is possible We saw this a couple a slides ago.

13. Graphical proof: the graphs seem to intersect at (2, -2). When x is smaller than 2, y1 is greater than y2 We have two functions: y 1, y 2. Find the range of x values (domain) so that y 1 > y 2 Scenario 1 Scenario 2 This is impossible