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

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