# Thread: some ideas in inequalities that i don't get.

1. ## some ideas in inequalities that i don't get.

Hello

http://im35.gulfup.com/GDxl1.png

since my mother language is not English, I find it difficult to understand some explanations. and i hope here i could find some help.

the above picture has a very strange way of solving inequalities !!

2. ## Re: some ideas in inequalities that i don't get.

Originally Posted by SHEvA9
Hello

http://im35.gulfup.com/GDxl1.png

since my mother language is not English, I find it difficult to understand some explanations. and i hope here i could find some help.

the above picture has a very strange way of solving inequalities !!
1. You are right: The explanation refers to the inequality

$(x-2)(x+3)>0$

2. Here are some (possible) steps to solve the given inequality:
• Re-write so you have only one quotient.
• Determine numerator and denominator so that the complete quotient is positive.

3. In detail:

$\frac{2x-3}{3x-5}\ge 3~\implies~\frac{2x-3}{3x-5}-3 \ge 0~\implies~\frac{-7x+12}{3x-5}\ge 0$

4. A quotient equals zero or is greater than zero if numerator and denominator have the same sign:

Case 1: Both are positive: (Be aware that the denominator can't be zero)

$-7x+12\ge 0~\wedge~3x-5 > 0~\implies~x \le \frac{12}7~\wedge~x > \frac53$

That means: $\frac53 < x \le \frac{12}7$

Case 2: I'll leave this one for you.

3. ## Re: some ideas in inequalities that i don't get.

thank you a lot
I agree with what you said
and i got case 2 as it is the same thing as case 1
the two factors are negative, the inequality will be reversed when you multiply by negative denominator then reverse it back after dividing by negative sign of the nominator, right?

one last thing, " 3,000 Solved Problems in Calculus " is a book i am trying to finish so that i could be ready to study Math in Canadian university, is it a good book? i mean not just in terms of the language i also want to improve my math skills, is the book useful for this purpose?

4. ## Re: some ideas in inequalities that i don't get.

Originally Posted by SHEvA9
thank you a lot
I agree with what you said
and i got case 2 as it is the same thing as case 1
the two factors are negative, the inequality will be reversed when you multiply by negative denominator then reverse it back after dividing by negative sign of the nominator, right?

one last thing, " 3,000 Solved Problems in Calculus " is a book i am trying to finish so that i could be ready to study Math in Canadian university, is it a good book? i mean not just in terms of the language i also want to improve my math skills, is the book useful for this purpose? Sorry, but I don't know this book. Remember I live in Germany. Look into the Member's list who's living in Canada and ask her/him by PM.
Honestly I don't understand your answer refering to the 2nd case. This is what I did:

$\frac{2x-3}{3x-5}\ge 3~\implies~\frac{2x-3}{3x-5}-3 \ge 0~\implies~\frac{-7x+12}{3x-5}\ge 0$

That's of course the same as in case 1.

Now both, the numerator and the denominator, are negative:

$-7x+12\le 0~\wedge~3x-5 < 0~\implies~x \ge \frac{12}7~\wedge~x < \frac53$

That means: $x\in \emptyset$

If you draw the graphs of
$f(x)=\frac{2x-3}{3x-5}$ and y = 3 and $x = \frac53$ you'll find the set of solutions in the small thick red bar.