Respuesta :
The number of tennis matches that she could have won before the weekend began is 0.
Before the weekend, the player had a win ratio of 0.500, which indicates that she had won exactly as many games as she had lost. Call this number x for now. She has so lost x matches in addition to the x matches she won before to the weekend.
She competes in three matches during the course of the weekend and loses one, bringing her overall win total (after the weekend) to x + 3 and her overall loss total (after the weekend) to x + 1. Consequently, her weekend victory ratio is
[tex]\frac{(x + 3)}{(x + 1)}[/tex]
= [tex]\frac{(3x + 3)}{(x + 1) }[/tex]
= [tex]\frac{3x }{x + 1 }[/tex]
= [tex]\frac{3}{\frac{1}{x+1} }[/tex]
given that this value is greater than 0.503,
Thus,
[tex]\frac{3}{\frac{1}{x+1} }[/tex] > 0.503
Simplifying,
[tex]\frac{3}{\frac{1}{x+1} }[/tex]> 0.503 - 1
= -0.497
Further,
[tex]\frac{3}{\frac{1}{x+1} }[/tex] > -0.497
[tex]\frac{3}{\frac{1}{x+1} }[/tex] is equivalent to 3x
This inequality can be written as
3x > -0.497
3x is positive, dividing on both sides by 3 we get
x > -0.497 / 3
Simplifying,
x > -0.166
The least value of x that meets this inequality is 0, as x, the number of matches won, must be a positive integer. Thus, the maximum number of games the player might have won prior to the start of the weekend is x = 0, and the response is 0.
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