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

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 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 : Examples of linear inequalities: 1.2. Dividing by  1 gives  1 and  2 BUT  1 is greater than  2 So, We know ( 1 is less than 2 ) 11 22 

Linear and Quadratic Inequalities Linear Inequalities These inequalities can be solved like linear equations EXCEPT that multiplying or dividing by a negative number reverses the inequality. Examples of linear inequalities: 1.2. Dividing by  1 gives  1 and  2 BUT  1 is greater than  2 So, The inequality has been reversed We know ( 1 is less than 2 ) Consider the numbers 1 and 2 :

Linear and Quadratic Inequalities 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 and Quadratic Inequalities Exercises Find the range of values of x satisfying the following linear inequalities: Solution: Solution: Either OrDivide by -4: so,

Linear and Quadratic 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

Linear and Quadratic Inequalities 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 or These represent 2 separate intervals and CANNOT be combined

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

Linear and Quadratic Inequalities  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

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

Linear and Quadratic Inequalities

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Linear and Quadratic Inequalities  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

Linear and Quadratic Inequalities 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

Linear and Quadratic 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

Linear and Quadratic Inequalities 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 or These represent 2 separate intervals and CANNOT be combined