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Re: Inequality questions [#permalink]
11 Feb 2008, 07:44

greatchap wrote:

Thanks for the solutions maratikus.

I understood the second and the third problem.

In the first one (4/x-2 < 2) what is keeping me confused is why cant we straightaway cross multiply and get answer. ?

When you solve a problem, you can replace it with equivalent problems and solve those. However, if you multiply both sides by x-2 the new problem is not going to be equivalent to the initial one because there are going to be two situations:

4 < 2*(x-2) if x-2 > 0 4 > 2*(x-2) if x-2 < 0

because when you multiply both sides by the same positive number, inequality sign doesn't change, and the sign changes when you multiply both sides by the same negative number (for example if (-2) < (-1) and you multiply both sides by (-1) it would be incorrect to say: (-1)*(-2) < (-1)*(-1) or 2 < 1)

In order to avoid this problem, I mutliply both sides by a positive number (x-2)^2 and keep the inequality sign unchanged. I hope that helps.

Re: Inequality questions [#permalink]
11 Feb 2008, 07:52

greatchap wrote:

Thanks for the solutions maratikus.

I understood the second and the third problem.

In the first one (4/x-2 < 2) what is keeping me confused is why cant we straightaway cross multiply and get answer. ?

because x is a variable and we have an inequality here. u cant cross multiply unles u know the sign of the variable. a negative flips the inequality sign _________________

You tried your best and you failed miserably. The lesson is 'never try'. -Homer Simpson

Re: Inequality questions [#permalink]
11 Feb 2008, 08:55

maratikus wrote:

greatchap wrote:

Thanks for the solutions maratikus. 4 < 2*(x-2) if x-2 > 0 4 > 2*(x-2) if x-2 < 0

because when you multiply both sides by the same positive number, inequality sign doesn't change, and the sign changes when you multiply

here is my question: when you solve both of those inequalities, you get that either x is less than 4, or that x is greater than 4. how do you get to the final answer of x<2 and x>4 ?

Re: Inequality questions [#permalink]
13 Feb 2008, 10:23

Expert's post

\(\frac{4}{(x-2)} < 2\)

my logic:

1. it is true for all negative (x-2) or for x<2 2. the function \(f=\frac{4}{(x-2)}\) is \(\infty\) at x close to 2 at the greater side. 3. When x increases from 2, the function f always decreases and equal to 2 at x=4. Therefore, f<2 at x>4 4. Combine two conditions: x<2 & x>4 _________________

Re: Inequality questions [#permalink]
13 Feb 2008, 22:50

Expert's post

pmenon wrote:

walker wrote:

\(\frac{4}{(x-2)} < 2\)

my logic:

1. it is true for all negative (x-2) or for x<2

how can you show this algebraically ?

im getting 4 >x....

1. the inequality is true when 4/(x-2) is negative. 2. 4/(x-2) is negative when (x-2) is negative 3. (x-2) is negative when x<2 (x-2<0 --> x<2) _________________

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