9: linear and quadratic inequalities © christine crisp “teach a level maths” vol. 1: as core...
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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
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