Does Absolute Value Affect Fraction Equality?

In summary, the expressions ##\frac{|x + 1|}{|x + 2|}## and ##\left|\frac{x + 1}{x + 2} \right|## are equivalent, as long as x is not equal to -2. This is because the absolute value function only affects the final result, not the individual components of the fraction. Therefore, the answer to the original question is yes.
  • #1
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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  • #2
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.
 
  • #3
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?
 
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  • #4
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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Related to Does Absolute Value Affect Fraction Equality?

1. What is absolute value?

Absolute value is a mathematical concept that represents the distance of a number from zero on a number line. It is always a positive value, regardless of the sign of the number.

2. How do you find the absolute value of a number?

To find the absolute value of a number, you simply remove the negative sign (if present) and keep the positive value. For example, the absolute value of -5 is 5.

3. What is the difference between absolute value and regular value?

The regular value of a number represents its numerical value, while the absolute value represents its distance from zero. Regular value can be positive or negative, while absolute value is always positive.

4. How is absolute value used in real life?

Absolute value is used in many real-life situations, such as calculating distances, determining the magnitude of a change, and finding the difference between two values. It is also used in solving equations and inequalities in math.

5. Can the absolute value of a number be negative?

No, the absolute value of a number is always positive or zero. This is because it represents the distance from zero, which is always positive. Negative numbers can only have a negative regular value, not an absolute value.

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