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Zorluk: KolayPositive and Negative Number Properties

If xx is a negative real number and yy is a positive real number, what is the value of x∣x∣+∣y∣y\frac{x}{|x|} + \frac{|y|}{y}?

Cevap: 0

Cevap

The value of the expression is 0.
For any negative number xx, the ratio x∣x∣\frac{x}{|x|} evaluates to −1-1 because ∣x∣=−x|x| = -x. For any positive number yy, the ratio ∣y∣y\frac{|y|}{y} evaluates to 11 because ∣y∣=y|y| = y. Summing −1-1 and 11 results in 00.

Adım Adım Çözüm

1
Evaluate the first term for a negative variable
-1
By definition of absolute value, if x<0x < 0, then ∣x∣=−x|x| = -x, making x∣x∣=x−x=−1\frac{x}{|x|} = \frac{x}{-x} = -1.
2
Evaluate the second term for a positive variable
1
If y>0y > 0, then ∣y∣=y|y| = y, making ∣y∣y=yy=1\frac{|y|}{y} = \frac{y}{y} = 1.
3
Sum the simplified values
0
Combining −1-1 and 11 yields −1+1=0-1 + 1 = 0.

Anahtar Kavram

Properties of Absolute Value and Signs of Numbers
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