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“Teach A Level Maths” Vol. 1: AS Core Modules 9: Linear and Quadratic Inequalities © Christine Crisp Linear and Quadratic Inequalities Module C1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Linear and Quadratic Inequalities Linear Inequalities Examples of linear inequalities: 1. 2. 4 3 x 8 x 3 x 12 These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality. Consider the numbers 1 and 2 : We know 1 2 ( 1 is less than 2 ) Dividing by 1 gives 1 and 2 BUT 1 is greater than 2 So, 1 2 2 1 0 1 2 Linear and Quadratic Inequalities Linear Inequalities Examples of linear inequalities: 1. 2. 4 3 x 8 x 3 x 12 These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality. Consider the numbers 1 and 2 : We know 1 2 ( 1 is less than 2 ) Dividing by 1 gives 1 and 2 BUT 1 is greater than 2 So, 1 2 The inequality has been reversed Linear and Quadratic Inequalities e.g.1. Find the values of x that satisfy the inequality 3 x 12 Solution: Divide by 3 x 4 e.g.2 Find the range of values of x that satisfy the inequality 4 3 x 8 x Solution: Collect the like terms Notice the change x 3x 8 4 from “less than” 4x 4 to “greater than” Divide by 4: x 1 Tips: Collecting the x-terms on the side which makes the coefficient positive avoids the need to divide by a negative number Substitute one value of x as a check on the answer Linear and Quadratic Inequalities Exercises Find the range of values of x satisfying the following linear inequalities: 1. 4x 1 2x 3 Solution: 4 x 2 x 3 1 2. 2 x 4 x 2 7 3x x 1 Solution: Either 7 1 x 3 x 8 4 x 2 x so, x 2 Or 4 x 8 Divide by -4: x 2 Linear and Quadratic Inequalities Quadratic Inequalities e.g.1 Find the range of values of x that satisfy x 2x 3 2 Solution: Method: ALWAYS use a sketch Rearrange to get zero on one side: x 2 2 x 3 0 Let f ( x ) x 2 2 x 3 and find the zeros of y f ( x ) x 2 2 x 3 0 ( x 1)( x 3) 0 x 1 or x 3 y x2 2x 3 x 2 2 x 3 is less than 0 below the x-axis The corresponding x values are between -3 and 1 3 x 1 Linear and Quadratic Inequalities e.g.2 Find the values of x that satisfy x 2 4 x 5 0 Solution: Find the zeros of f ( x ) where f ( x ) x 2 4 x 5 x 2 4 x 5 0 ( x 5)( x 1) 0 x5 or x 1 x 2 4 x 5 is greater than or equal to 0 above the x-axis There are 2 sets of values of x x 1 or x 5 These represent 2 separate intervals and CANNOT be combined yy x 22 4 x 5 Linear and Quadratic Inequalities e.g.3 Find the values of x that satisfy 4 x x 2 0 Solution: Find the zeros of f ( x ) where f ( x ) 4 x x 2 4x x2 0 x (4 x ) 0 This quadratic has a common factor, x y 4 x x 2 x 0 or x 4 than sketching this0quadratic as the coefficient of 4Be x careful x 2 is greater 2 negative. x aboveisthe x-axis The quadratic is “upside down”. 0 x4 Linear and Quadratic Inequalities SUMMARY Linear inequalities Solve as for linear equations BUT • Keep the inequality sign throughout the working • If multiplying or dividing by a negative number, reverse the inequality Quadratic ( or other ) Inequalities • rearrange to get zero on one side, find the zeros and sketch the function • • Use the sketch to find the x-values satisfying the inequality Don’t attempt to combine inequalities that describe 2 or more separate intervals Linear and Quadratic Inequalities Exercise 1. Find the values of x that satisfy f ( x ) 0 where f ( x ) x 2 7 x 10 Solution: x 2 7 x 10 0 ( x 5)( x 2) 0 x 5 or x2 y x 2 7 x 10 x 2 7 x 10 is greater than or equal to 0 above the x-axis There are 2 sets of values of x which cannot be combined x 2 or x 5 Linear and Quadratic Inequalities The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet. Linear and Quadratic Inequalities SUMMARY Linear inequalities Solve as for linear equations BUT • Keep the inequality sign throughout the working • If multiplying or dividing by a negative number, reverse the inequality Quadratic ( or other ) Inequalities • rearrange to get zero on one side, find the zeros and sketch the function • • Use the sketch to find the x-values satisfying the inequality Don’t attempt to combine inequalities that describe 2 or more separate intervals Linear and Quadratic Inequalities Linear Inequalities e.g.1. Find the values of that satisfy the inequality 3 x 12 Solution: Divide by 3 x 4 e.g.2 Find the range of values of x that satisfy the inequality 4 3 x 8 x Solution: Collect the like terms Notice the change x 3x 8 4 from “less than” 4x 4 to “greater than” Divide by -4: x 1 Tips: Collecting the x-terms on the side which makes the coefficient positive avoids the need to divide by a negative number Substitute one value of x as a check on the answer Linear and Quadratic Inequalities Quadratic Inequalities e.g.1 Find the range of values of x that satisfy x 2x 3 2 Solution: Method: ALWAYS use a sketch Rearrange to get zero on one side: x 2 2 x 3 0 Let f ( x ) x 2 2 x 3 and find the zeros of y f ( x ) x 2 2 x 3 0 ( x 1)( x 3) 0 x 1 or x 3 y x2 2x 3 x 2 2 x 3 is less than 0 below the x-axis The corresponding x values are between -3 and 1 3 x 1 Linear and Quadratic Inequalities e.g.2 Find the values of x that satisfy x 2 4 x 5 0 Solution: Find the zeros of f ( x ) where f ( x ) x 2 4 x 5 x 2 4 x 5 0 ( x 5)( x 1) 0 x5 or x 1 y x2 4x 5 x 2 4 x 5 is greater than or equal to 0 above the x-axis There are 2 sets of values of x x 1 or x 5 These represent 2 separate intervals and CANNOT be combined