Pierādi, ka izteiksmes (1/(y-16) - 1/(y-13)) : (y-13)/(y-16) vērtība ir pozitīva visiem y, ar kuriem izteiksme ir definēta. Risini kopā ar uzdevumi.lv! Papildini ar skaitļiem!

Proof that the expression (1/(y-16) - 1/(y-13)) : (y-13)/(y-16) is positive for all y for which the expression is defined. Solve together with uzdevumi.lv! Fill in the numbers!

Solution:

1. Simplify the expression inside the parentheses:
   • Find a common denominator: $$(y-16)(y-13)$$
   • Rewrite the expression: $$\frac{(y-13) - (y-16)}{(y-16)(y-13)}$$
   • Simplify the numerator: $$\frac{y - 13 - y + 16}{(y-16)(y-13)} = \frac{3}{(y-16)(y-13)}$$
2. Divide by the second fraction:
   • Dividing by a fraction is the same as multiplying by its reciprocal.
   • So, the expression becomes: $$\frac{3}{(y-16)(y-13)} \times \frac{y-16}{y-13}$$
3. Simplify the resulting expression:
   • Cancel out the common term $$(y-16)$$: $$\frac{3}{1} \times \frac{1}{(y-13)(y-13)} = \frac{3}{(y-13)^2}$$
4. Analyze the result:
   • The numerator is 3, which is a positive constant.
   • The denominator is $$(y-13)^2$$. For any real value of y (except y=13, where the original expression is undefined), $$(y-13)^2$$ will be positive because it is a square of a non-zero number.
   • Therefore, a positive number divided by a positive number is always positive.

The filled-in numbers are:

... = $$\frac{\boxed{3}}{(y-16)(y\boxed{-13})} : \frac{y-13}{y-16}$$

= $$\frac{\boxed{3}}{\boxed{(y-13)^2}}$$

Tā kā daļas skaitītājs un saucējs ir pozitīvi visiem y, ar kuriem izteiksme ir definēta, var secināt, ka dotās izteiksmes vērtība ir pozitīva visiem y, ar kuriem izteiksme ir definēta.

Final Answer: The expression $$\frac{3}{(y-13)^2}$$ is positive for all $$y$$ where it is defined, because the numerator is positive and the denominator is a square of a non-zero real number, making it positive.