B Does Absolute Value Affect Fraction Equality?

askor
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Is it correct that ##\frac{|x + 1|}{|x + 2|}## equal to ##\left|\frac{x + 1}{x + 2} \right|##?

Please explain, I don't understand.

Thank you
 
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Pick an x but not x=-2 and test it:

I pick x = 10

|x+1| / |x+2| = |11| / |12| = 11 / 12 = | 11/12 | = | (x+1) / (x+2) |

then pick x=-10

| x+1| / |x+2| = |-9| / |-8| = 9 / 8 = | 9/8 | = | -9/-8 | = | (x+1) / (x+2) |

Try other values for x and then decide if it is true or not.
 
askor said:
Is it correct that ##\frac{|x + 1|}{|x + 2|}## equal to ##\left|\frac{x + 1}{x + 2} \right|##?
Yes, the two expressions are identically equal.
askor said:
Please explain, I don't understand.
Think about the expressions x + 1 and x + 2. Each of them is negative, zero, or positive, depending on the value of x. Now, as long as ##x \ne -2##, ##\frac{x +1}{x + 2}## will have some value. Does it matter whether we take the absolute values of the numerator and denominator separately, or evaluate the fraction and then take its absolute value?
 
askor said:
Is it correct that ##\frac{|x + 1|}{|x + 2|}## equal to ##\left|\frac{x + 1}{x + 2} \right|##?
Absolutely!

It's only ##|xy| = |x||y|## in disguise.
 
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