Does Absolute Value Affect Fraction Equality?

[askor]

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

 

[jedishrfu]

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.

 

[Mark44]

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.

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?

 

[PeroK]

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.