Inequalities

How do u do inequalities like:

1) Find the set of values of x for which :
a) 2x^2-11x+5<0
b) both 6x-7<2x+3 and 2x^2-11x+5<0

2) Find algebraically the set of values of x for which (2x-3)(x+2)>(3(x-2)

3) Use algebra to find the set of values of x for which x(x-5)>36. Using your answer to part a, find the set of values of y for which y^2(y^2-5)>36.

4) And...find, as surds, the roots of the equation 2(x+1)(x-4)-(x-2)^2=0. Hence find the set of values of x for which 2(x+1)(x-4)-(x-2)^2>0.

Well, im *really* hoping that someone can help me!!!!!!! :redface:

Reply 1

mathz

8

*girlie*

How do u do inequalities like:

1) Find the set of values of x for which :
a) 2x^2-11x+5<0
b) both 6x-7<2x+3 and 2x^2-11x+5<0

2) Find algebraically the set of values of x for which (2x-3)(x+2)>(3(x-2)

3) Use algebra to find the set of values of x for which x(x-5)>36. Using your answer to part a, find the set of values of y for which y^2(y^2-5)>36.

4) And...find, as surds, the roots of the equation 2(x+1)(x-4)-(x-2)^2=0. Hence find the set of values of x for which 2(x+1)(x-4)-(x-2)^2>0.

Well, im *really* hoping that someone can help me!!!!!!! :redface:

2x^2-11x+5<0
gives (2x-1)(x-5)<0
so 2x-1>0 and x-5<0 which gives x>1/2 and x<5
or 2x-1<0 and x-5>0 which gives x<1/2 and x>5 which is impossible
so 1/2<x<5

b) both 6x-7<2x+3 and 2x^2-11x+5<0
6x-7<2x+3 gives 4x<10 so x<5/2
2x^2-11x+5<0 gives 1/2<x<5
so if x satisfies both must have 1/2<x<5/2

c) (2x-3)(x+2)>3(x-2)
2x^2+x-6>3x-6
2x^2-2x>0
x(x-1)>0
so x>0 and x-1>0 gives x>1
or x<0 and x-1<0 gives x<0
so x<0 or x>2

d) x(x-5)>36.
x^2-5x-36>0
(x-9)(x+4)>0
x-9>0 and x+4>0 gives x>9
x-9<0 and x+4<0 gives x<-4
so x<-4 or x>9
y^2(y^2-5)>36 let y^2=x and use above to get
so y^2>9 so y>3 or y<-3
y^2<-4 no solution

so y<-3 or y>3

e 2(x+1)(x-4)-(x-2)^2=0.
2x^2-6x-8-{x^2-2x+4}=0
x^2-4x-4=0
x=4+-sqrt(16+16)/2
x=2+2sqrt(2)
x=2-2sqrt(2)
2(x+1)(x-4)-(x-2)^2>0.
so x^2-4x-4>0
(x-2+2sqrt(2))(x-(2-2sqrt(2)))>0
so x>2+2sqrt(2)) and x>2-2sqrt(2) so x>2+2sqrt(2)
or x<2+2sqrt(2)) and x<2-2sqrt(2) so x<2-2sqrt(2)
so x>2+2sqrt(2) or x<2-2sqrt(2)

Reply 2

*girlie*

OP

5

oo thanks a lot, i really appreciate it!! :smile:

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