অসমতার যোগ, বিয়োগ, গুন, ভাগ (BOMC-2011)

[sm.joty]

অসমতার যোগ ,বিয়োগ, গুন, ভাগের বিষয়টা ভালো ভাবে জানা দরকার। যেমন বিয়গের ক্ষেত্রে, Ex-1.7 এ বলা আছে,\[0\leqslant a\leqslant b\leqslant 1\]
এখান থেকে বলা যায়,\[ a\leqslant 1\] এবং \[ b\leqslant 1\]
তাহলে,\[ b-a \leqslant 0\]
অর্থাৎ, \[b\leq a\]
কিন্তু এটাতো শর্ত বিরধি...।তাহলে আসল বিষয়টা কী ?
আবার গুনের ক্ষেত্রেও\[a>b, x>y,m>n \]
হলে কি সবসময় এটা সত্যি যে,\[axm>byn\]

[FahimFerdous]

Not always. But if they're positive, then it's true.

[sourav das]

Let us say "There is no subtraction in inequality, only addition". Now $a \leq 1$ . Multiply $-1$.Then $-a \geq -1$
First confusion is over (I hope). Now again we can multiply only when every number is positive. So not every time it is true.

[*Mahi*]

Let us say "There is no subtraction in inequality, only addition". Now $a \leq 1$ . Multiply $-1$.Then $-a \geq -1$
First confusion is over (I hope). Now again we can multiply only when every number is positive. So not every time it is true.

And there is no dividing too, just multiplication, sometimes with the multiplicative inverse.

[sourav das]

Huuu. That's true.

[MATHPRITOM]

Not always. But if they're positive, then it's true.

i don't understand what u meant by the sentence.plz,explain details.

[nafistiham]

Not always. But if they're positive, then it's true.

i don't understand what u meant by the sentence.plz,explain details.

i think he said if $a,x,m,b,y,n$ all are positive,then $axm>byn$

[sm.joty]

That means I can't apply division in inequality.

[sourav das]

@Joti, Of course, you can. But you have to justify if it is correct. That is why first we multiply inversely instead of division. Same idea goes for subtraction and power.

[*Mahi*]

You can divide using inverses , though.
Let $a>b$ and $c>d$ , $a,b,c,d$ positive. Then of course $\frac 1 d > \frac 1 c$ , and you can multiply them to get $\frac a d >\frac b c$