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 moredumbimpossible May 19th, 2018 07:05 PM

http://i.imgur.com/o5CLFI7.jpg

Could you please help me to solve these inequations..? I have many more and won't be able to solve them all :eek::eek:.. I need help.

 romsek May 19th, 2018 07:29 PM

If there are specific questions you have regarding specific problems we can possibly help.

We don't do your homework for you.

 moredumbimpossible May 19th, 2018 08:17 PM

Quote:
 Originally Posted by romsek (Post 594681) If there are specific questions you have regarding specific problems we can possibly help. We don't do your homework for you.
The homework is around 100 problems, I only posted 7.

 romsek May 19th, 2018 08:22 PM

Quote:
 Originally Posted by moredumbimpossible (Post 594682) The homework is around 100 problems, I only posted 7.
These are all pretty straightforward linear inequalities.

In all cases you can do the algebra to reduce it to a simple statement of inequality.

For example the first one

\begin{align*} &\dfrac{a+2}{4} \leq \dfrac{a-1}{3}; \\ \\ &3(a+2) \leq 4(a-1); \\ \\ &3a+6 \leq 4a - 4; \\ \\ &10 \leq a \end{align*}

None of the rest of them are any more complicated.

 JeffM1 May 19th, 2018 08:53 PM

You can always deal with an inequation by solving the corresponding equation. Example.

$\text {In what interval or intervals is } \dfrac{4x - 3}{3} < \dfrac{2x + 4}{4}.$

Start by solving

$\dfrac{4x - 3}{3} = \dfrac{2x + 4}{4} \implies 16x - 12 = 6x + 12 \implies$

$10x = 24 \implies x = 2.4.$

Is 2.4 in the interval you want? No because we are looking for less than, not less than or equal. If we were looking for less than or equal, 2.4 would be in an interval.

You have now divided the number line into two parts, x < 2.4 and x > 2.4. Pick a number in each part and test.

Well 2 is less than 2.4.

$\dfrac{4 * 2 - 3}{3} = \dfrac{5}{3} = \dfrac{20}{12}.$

$\dfrac{2 * 2 + 4}{4} = \dfrac{8}{4} = \dfrac{24}{12}.$

$\dfrac{20}{12} < \dfrac{24}{12}.$

So numbers less than 2.4 are in the interval you want.

And 3 is greater than 2.4.

$\dfrac{4 * 3 - 3}{3} = \dfrac{9}{3} = \dfrac{36}{12}.$

$\dfrac{2 * 3 + 4}{4} = \dfrac{10}{4} = \dfrac{30}{36}.$

$\dfrac{36}{12} > \dfrac{30}{12}.$

So numbers greater than 2.4 are NOT in the interval you want.

Thus the answer is $(-\ \infty,\ 2.4).$

CAUTION

We had one point of equality in this problem so we divided the number line into 2 parts, and we had to test each part. But if we had two points of equality, we would divide the number line into three parts and would need to test each part. If we have n points of equality, we divide the number line into n + 1 parts and must test a number in each part.

EDIT:

Alternatively, you can try to solve a linear inequality directly. In my example,

$\dfrac{4x - 3}{3} < \dfrac{2x + 4}{4} \implies \dfrac{12(4x - 3)}{3} < \dfrac{12(2x + 4)}{4} \implies$

$16x - 12 < 6x + 12 \implies 10x < 24 \implies x < 2.4.$

That is quicker, but it gets tricky for more complex inequalities.

 romsek May 19th, 2018 10:13 PM

I ordinarily would never comment on someone else's method but ... really?

$\dfrac{4x-3}{3} < \dfrac{2x+4}{4}$

$4(4x-3) < 3(2x+4)$

$16x-12 < 6x + 12$

$10x < 24$

$x < \dfrac{24}{10} = \dfrac{12}{5}$

sometimes it pays to just keep it simple.

 moredumbimpossible May 20th, 2018 09:05 AM

Thanks that doesn't look that hard. Maybe you can explain how to solve notable products like this. This is the last one, I swear...

:cool:http://i.imgur.com/UL46iTT.jpg

 romsek May 20th, 2018 10:19 AM

There's nothing to solve here. These are just expressions that I guess they want you to multiply out.

You can do this; you're just lazy. Get to it.

 JeffM1 May 20th, 2018 10:27 AM

Romsek will probably say I am being too complex. You can solve these particular problems directly and easily using basic algebra and the laws of exponents, but you can also use a method that is particularly helpful for more complex cases. Let's take the last problem as an example.

Direct method.

$\left ( \dfrac{x^2}{2} - \dfrac{x}{3} \right ) \left ( \dfrac{x^2}{2} + \dfrac{x}{3} \right ) =$

$\left ( \dfrac{3x^2}{3 * 2} - \dfrac{2x}{2 * 3} \right ) \left ( \dfrac{3x^2}{3 * 2} + \dfrac{2x}{2 * 3} \right ) =$

$\dfrac{3x^2 - 2x}{6} * \dfrac{3x^2 + 2x}{6} =$

$\dfrac{9x^4 + 6x^3 - 6x^3 - 4x^2}{36} = \dfrac{9x^4 - 4x^2}{36} = \dfrac{9x^4}{36} - \dfrac{4x^2}{36} = \dfrac{x^4}{4} - \dfrac{x^2}{9}.$

Basic algebra.

Or you can do this, formally or in your head.

$u = \dfrac{x^2}{2} \text { and } v = \dfrac{x}{3}.$

$\therefore \left ( \dfrac{x^2}{2} - \dfrac{x}{3} \right ) \left ( \dfrac{x^2}{2} + \dfrac{x}{3} \right ) =$

$( u - v)(u + v) = u^2 - v^2 =$

$\left ( \dfrac{x^2}{2} \right )^2 - \left ( \dfrac{x}{3} \right )^2 = \dfrac{x^4}{4} - \dfrac{x^2}{9}.$

There are sometimes many correct ways to attack a problem.

 moredumbimpossible May 21st, 2018 10:07 AM

Quote:
 Originally Posted by romsek (Post 594701) There's nothing to solve here. These are just expressions that I guess they want you to multiply out. You can do this; you're just lazy. Get to it.
I checked on a book, and I solved those applying notable products.
(a+b)(a-b)...
It seems like you're the only one here so I'll assume that you are some sort of moderator too..
Please feel free to close this useless account. Bye..

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