inequality of variables without knowing their sign
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I would like to manipulate an inequality when the said inequality is in terms of variables and I do not know the signs of the variables.
For example, $x_1<x_2$ and I would like to divide both sides by $x_1$.
If $x_1<0 $, $x_1<x_2 implies 1 >frac{x_2}{x_1} $.
If $x_1>0 $, $x_1<x_2 implies 1 < frac{x_2}{x_1} $.
But, without haivng this knowledge, do I always have to split the inequality into two separate cases, as I attempt to simplify my expression?
inequality
asked Jan 10 at 23:05
BlackwidowBlackwidow
1
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add a comment |
$begingroup$
I would like to manipulate an inequality when the said inequality is in terms of variables and I do not know the signs of the variables.
For example, $x_1<x_2$ and I would like to divide both sides by $x_1$.
If $x_1<0 $, $x_1<x_2 implies 1 >frac{x_2}{x_1} $.
If $x_1>0 $, $x_1<x_2 implies 1 < frac{x_2}{x_1} $.
But, without haivng this knowledge, do I always have to split the inequality into two separate cases, as I attempt to simplify my expression?
inequality
asked Jan 10 at 23:05
BlackwidowBlackwidow
1
$endgroup$
$begingroup$
Yes, indeed you do.
$endgroup$
– user3482749
Jan 10 at 23:08
add a comment |
$begingroup$
I would like to manipulate an inequality when the said inequality is in terms of variables and I do not know the signs of the variables.
For example, $x_1<x_2$ and I would like to divide both sides by $x_1$.
If $x_1<0 $, $x_1<x_2 implies 1 >frac{x_2}{x_1} $.
If $x_1>0 $, $x_1<x_2 implies 1 < frac{x_2}{x_1} $.
But, without haivng this knowledge, do I always have to split the inequality into two separate cases, as I attempt to simplify my expression?
inequality
asked Jan 10 at 23:05
BlackwidowBlackwidow
1
$endgroup$
I would like to manipulate an inequality when the said inequality is in terms of variables and I do not know the signs of the variables.
For example, $x_1<x_2$ and I would like to divide both sides by $x_1$.
If $x_1<0 $, $x_1<x_2 implies 1 >frac{x_2}{x_1} $.
If $x_1>0 $, $x_1<x_2 implies 1 < frac{x_2}{x_1} $.
But, without haivng this knowledge, do I always have to split the inequality into two separate cases, as I attempt to simplify my expression?
inequality
inequality
asked Jan 10 at 23:05
BlackwidowBlackwidow
1
asked Jan 10 at 23:05
BlackwidowBlackwidow
1
asked Jan 10 at 23:05
BlackwidowBlackwidow
1
asked Jan 10 at 23:05
BlackwidowBlackwidow
1
asked Jan 10 at 23:05
BlackwidowBlackwidow
1
1
$begingroup$
Yes, indeed you do.
$endgroup$
– user3482749
Jan 10 at 23:08
add a comment |
$begingroup$
Yes, indeed you do.
$endgroup$
– user3482749
Jan 10 at 23:08
$begingroup$
Yes, indeed you do.
$endgroup$
– user3482749
Jan 10 at 23:08
$begingroup$
Yes, indeed you do.
$endgroup$
– user3482749
Jan 10 at 23:08
add a comment |
1 Answer
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$begingroup$
Yes, you are correct. You need to consider both cases.
answered Jan 11 at 0:45
Ekesh KumarEkesh Kumar
6046
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add a comment |
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1 Answer
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1 Answer
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$begingroup$
Yes, you are correct. You need to consider both cases.
answered Jan 11 at 0:45
Ekesh KumarEkesh Kumar
6046
$endgroup$
add a comment |
$begingroup$
Yes, you are correct. You need to consider both cases.
answered Jan 11 at 0:45
Ekesh KumarEkesh Kumar
6046
$endgroup$
add a comment |
$begingroup$
Yes, you are correct. You need to consider both cases.
answered Jan 11 at 0:45
Ekesh KumarEkesh Kumar
6046
$endgroup$
Yes, you are correct. You need to consider both cases.
answered Jan 11 at 0:45
Ekesh KumarEkesh Kumar
6046
answered Jan 11 at 0:45
Ekesh KumarEkesh Kumar
6046
answered Jan 11 at 0:45
Ekesh KumarEkesh Kumar
6046
answered Jan 11 at 0:45
Ekesh KumarEkesh Kumar
6046
6046
add a comment |
add a comment |
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Yes, indeed you do.
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– user3482749
Jan 10 at 23:08