Download - 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules
![Page 1: 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules](https://reader036.vdocuments.site/reader036/viewer/2022083009/5697bf791a28abf838c8283b/html5/thumbnails/1.jpg)
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
![Page 2: 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules](https://reader036.vdocuments.site/reader036/viewer/2022083009/5697bf791a28abf838c8283b/html5/thumbnails/2.jpg)
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
![Page 3: 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules](https://reader036.vdocuments.site/reader036/viewer/2022083009/5697bf791a28abf838c8283b/html5/thumbnails/3.jpg)
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
![Page 4: 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules](https://reader036.vdocuments.site/reader036/viewer/2022083009/5697bf791a28abf838c8283b/html5/thumbnails/4.jpg)
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
![Page 5: 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules](https://reader036.vdocuments.site/reader036/viewer/2022083009/5697bf791a28abf838c8283b/html5/thumbnails/5.jpg)
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
![Page 6: 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules](https://reader036.vdocuments.site/reader036/viewer/2022083009/5697bf791a28abf838c8283b/html5/thumbnails/6.jpg)
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:
![Page 7: 9: Linear and Quadratic Inequalities © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules](https://reader036.vdocuments.site/reader036/viewer/2022083009/5697bf791a28abf838c8283b/html5/thumbnails/7.jpg)
Linear and Quadratic Inequalities