Is |a|-|b|<|a-b|?
1) b^a<0
2) |b|>|a|
OA: B
Can I square both sides in question stem as follows?
Is (|a|-|b|)^2<(|a-b|)^2
a^2 -2 |a||b| + b^2 < a^2 -2 ab+ b^2
-2 |a||b| < -2 ab.......divide by -2
Is |a||b| > ab
|a|-|b|<|a-b|?
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No, you cannot do that - in general, if you have an inequality, if one side of the inequality might be negative, it is not correct to square both sides. For example, if you start with this inequality, which is clearly true:Mo2men wrote: Can I square both sides in question stem as follows?
-3 < 2
then if you square both sides, you'll get something that is false (9 is not less than 4). So squaring both sides of an inequality is mathematically illegal (except when you know both sides are positive). Here, the right side of the inequality is certainly positive, but the left side, |a| - |b|, might be negative. So squaring is not correct.
If you know how to interpret absolute values as distances (I don't have time to explain that method now, but I've posted about it several times in the past on this forum, so you might find an explanation with a search), you can answer this question fairly quickly. The answer to the question is usually "yes", and is only "no" when b is in between a and zero. In that case, so if either a < b < 0 or 0 < b < a, the left and right sides of the inequality in the question are equal. Since Statement 2 tells us b is not in between a and zero, it is sufficient. Statement 1 is a bit problematic (the expression b^a can be undefined if a is a non-integer) but it only tells us b is negative, and tells us nothing important about where a is, so Statement 1 is not sufficient.
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Thanks Ian for your help and support.Ian Stewart wrote:No, you cannot do that - in general, if you have an inequality, if one side of the inequality might be negative, it is not correct to square both sides. For example, if you start with this inequality, which is clearly true:Mo2men wrote: Can I square both sides in question stem as follows?
-3 < 2
then if you square both sides, you'll get something that is false (9 is not less than 4). So squaring both sides of an inequality is mathematically illegal (except when you know both sides are positive). Here, the right side of the inequality is certainly positive, but the left side, |a| - |b|, might be negative. So squaring is not correct.
If you know how to interpret absolute values as distances (I don't have time to explain that method now, but I've posted about it several times in the past on this forum, so you might find an explanation with a search), you can answer this question fairly quickly. The answer to the question is usually "yes", and is only "no" when b is in between a and zero. In that case, so if either a < b < 0 or 0 < b < a, the left and right sides of the inequality in the question are equal. Since Statement 2 tells us b is not in between a and zero, it is sufficient. Statement 1 is a bit problematic (the expression b^a can be undefined if a is a non-integer) but it only tells us b is negative, and tells us nothing important about where a is, so Statement 1 is not sufficient.
Can you please suggest one of your posts the explain the absolute values as distances. The search does not help a lot to get ant post for you.
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I'm sure there are others, but on a quick search I found these:Mo2men wrote: Can you please suggest one of your posts the explain the absolute values as distances. The search does not help a lot to get ant post for you.
https://www.beatthegmat.com/mod-problem ... html#85748
https://www.beatthegmat.com/ds-t17361.html
https://www.beatthegmat.com/absolute-va ... 19267.html
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