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Inequality question

Q. Solve the inequality |x + 3a| > 2|x − 2a|, where a is a positive constant.

I managed to turn it into the quadratic form, 3x^2 - 22ax + 7a^2 > 0, but am not able to find the x values. Please help.
Original post by Maemi
Q. Solve the inequality |x + 3a| > 2|x − 2a|, where a is a positive constant.

I managed to turn it into the quadratic form, 3x^2 - 22ax + 7a^2 > 0, but am not able to find the x values. Please help.


You could always use a sketch to help you work out the inequality without having to resort to the quadratic. :smile:

Hint

(edited 10 years ago)
Reply 2
Avatar for Maemi
Maemi
OP
3
Original post by Khallil
You could always use a sketch to help you work out the inequality without having to resort to the quadratic. :smile:


Ok, will try that. Thanx
Original post by Khallil
You could always use a sketch to help you work out the inequality without having to resort to the quadratic. :smile:

Hint



Your quadratic factorises quite easily.
Original post by Maemi

I managed to turn it into the quadratic form, 3x^2 - 22ax + 7a^2 > 0, but am not able to find the x values. Please help.


Should be 3x^2 - 22ax + 7a^2 < 0
Reply 5
Avatar for DFranklin
DFranklin
18
Original post by Maemi
Q. Solve the inequality |x + 3a| > 2|x − 2a|, where a is a positive constant.

I managed to turn it into the quadratic form, 3x^2 - 22ax + 7a^2 > 0, but am not able to find the x values. Please help.
Re: solving the quadratic...

Write y = x/a, then 3x^2 - 22ax + 7a^2 becomes a^2(3y^2 - 22y + 7). You can either factorize this by inspection or use the quadratic formula.
Reply 6
Avatar for ztibor
ztibor
10
Original post by Maemi
Q. Solve the inequality |x + 3a| > 2|x − 2a|, where a is a positive constant.

I managed to turn it into the quadratic form, 3x^2 - 22ax + 7a^2 > 0, but am not able to find the x values. Please help.


Without quadratic: you have 3 cases
1) x>=2a
x+3a>2(x−2a)→x<7a→2a<=x<7ax+3a>2(x-2a) \rightarrow x<7a \rightarrow 2a<=x<7a

2) -3a<=x<2a, then
x+3a>2(2a−x)→3x>a→x>a3→a3<x<2ax+3a>2(2a-x) \rightarrow 3x>a \rightarrow x>\frac{a}{3} \rightarrow \frac{a}{3}<x<2a

3) x<-3a
−x−3a>2(2a−x)→x>7a-x-3a>2(2a-x) \rightarrow x>7a so no solution here

Summarizing
a3<x<7a\frac{a}{3}<x<7a
(edited 10 years ago)

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