9: Linear and 9: Linear and Quadratic InequalitiesQuadratic Inequalities

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

Linear and Quadratic Inequalities

322 xxy

Quadratic Inequalities

Solution:

e.g.1 Find the range of values of x that satisfy 322 xx

Rearrange to get zero on one side:

0322 xx

0322 xx 0)3)(1( xx

1 x or 3x

322 xx is less than 0 below the x-axis

13 xThe corresponding x values are between -3 and 1

Let and find the zeros of 32)( 2 xxxf )(xfy

Method: ALWAYS use a sketch

Linear and Quadratic Inequalities

542 xxy 542 xxy

Solution:e.g.2 Find the values of x that satisfy 0542 xx

0542 xx 0)1)(5( xx

5 x or 1x

1 x

There are 2 sets of values of x

Find the zeros of where )(xf 54)( 2 xxxf

542 xx is greater than orequal to 0 above the x-

axis

5xor

These represent 2 separate intervals and CANNOT be combined

Linear and Quadratic Inequalities

24 xxy

Solution:e.g.3 Find the values of x that satisfy 04 2 xx

04 2 xx

0)4( xx

40 x

Find the zeros of where )(xf 24)( xxxf

24 xx is greater than 0above the x-

axis

This quadratic has a common factor, x

or 4x0x

24 xxy

Be careful sketching this quadratic as the coefficient of is negative. The quadratic is “upside down”.2x

Linear and Quadratic Inequalities

Linear inequalitiesSolve 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

1072 xxy 1072 xxy

Exercise

01072 xx 0)2)(5( xx

5 x or 2x

2 x

There are 2 sets of values of x which cannot be combined

1072 xx is greater thanor equal to 0 above

the x-axis

5xor

1. Find the values of x that satisfy where 107)( 2 xxxf

0)( xf

Solution:

Linear and Quadratic Inequalities