“Teach A Level Maths” Vol. 1: AS Core Modules 9: Linear and Quadratic Inequalities © Christine Crisp

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"

Examples of linear inequalities: 1. 2. 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 is less than 2 ) Dividing by -1 gives -1 and -2 BUT -1 is greater than -2 So, -2 -1 1 2

Linear Inequalities Examples of linear inequalities: 1. 2. 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 is less than 2 ) Dividing by -1 gives -1 and -2 BUT -1 is greater than -2 So, The inequality has been reversed

e.g.1. Find the values of that satisfy the inequality Solution: Divide by 3 e.g.2 Find the range of values of x that satisfy the inequality Solution: Collect the like terms Notice the change from “less than” to “greater than” Divide by -4: 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

Exercises Find the range of values of x satisfying the following linear inequalities: 1. Solution: 2. Solution: Either so, Or Divide by -4:

Quadratic Inequalities e.g.1 Find the range of values of x that satisfy Solution: Method: ALWAYS use a sketch Rearrange to get zero on one side: Let and find the zeros of or is less than 0 below the x-axis The corresponding x values are between -3 and 1

These represent 2 separate intervals and CANNOT be combined e.g.2 Find the values of x that satisfy Solution: Find the zeros of where or is greater than or equal to 0 above the x-axis There are 2 sets of values of x or These represent 2 separate intervals and CANNOT be combined

e.g.3 Find the values of x that satisfy Solution: Find the zeros of where This quadratic has a common factor, x or is greater than 0 above the x-axis Be careful sketching this quadratic as the coefficient of is negative. The quadratic is “upside down”.

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

Exercise 1. Find the values of x that satisfy where Solution: or is greater than or equal to 0 above the x-axis There are 2 sets of values of x which cannot be combined or

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 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 SUMMARY

e.g.1. Find the values of that satisfy the inequality Divide by -4: Solution: Divide by 3 e.g.2 Find the range of values of x that satisfy the inequality Solution: Collect the like terms Notice the change from “less than” to “greater than” 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 Tips: Linear Inequalities

Quadratic Inequalities Solution: e.g.1 Find the range of values of x that satisfy Rearrange to get zero on one side: or is less than 0 below the x-axis The corresponding x values are between -3 and 1 Let and find the zeros of Method: ALWAYS use a sketch

These represent 2 separate intervals and CANNOT be combined Solution: e.g.2 Find the values of x that satisfy or There are 2 sets of values of x Find the zeros of where is greater than or equal to 0 above the x-axis These represent 2 separate intervals and CANNOT be combined