![\frac{3x}{x-4} - \frac{x+3}{x-4} = \frac{2x+7}{x^2-16}](https://tex.z-dn.net/?f=%5Cfrac%7B3x%7D%7Bx-4%7D%20%20-%20%5Cfrac%7Bx%2B3%7D%7Bx-4%7D%20%3D%20%5Cfrac%7B2x%2B7%7D%7Bx%5E2-16%7D)
We factor the denominators
Factor x^2 - 16
x^2 - 4^2
We use a^2 - b^2 = (a+b)(a-b)
so x^2 - 4^2 = (x+4)(x-4)
Replace it in the given equation
![\frac{3x}{x-4} - \frac{x+3}{x-4} = \frac{2x+7}{(x+4)(x-4)}](https://tex.z-dn.net/?f=%5Cfrac%7B3x%7D%7Bx-4%7D%20%20-%20%5Cfrac%7Bx%2B3%7D%7Bx-4%7D%20%3D%20%5Cfrac%7B2x%2B7%7D%7B%28x%2B4%29%28x-4%29%7D)
Excluded values are the values that makes the denominator 0
we have (x-4) and (x+4) in the denominator
We set the denominator =0 and solve for x
x-4 =0
Add 4 on both sides
x= 4
x+4=0
subtract 4 onboth side
so x= -4
Excluded values are x=-4 and x=4
Answer: The picture's dimensions will be 10"x15".
Step-by-step explanation:
Length: 4 2 10
Width: 6 3 15
You can tell because 15 isn't divisible by 6, but it is divisible by 3, which goes into 6.
Divide 6 by 2.
6/2=3
To keep the ratio, we must also divide 4 by 2.
4/2=2
Then, multiply 3 by 5 for the width.
3*5=15
You must also multiply 2 by 5 to get the length.
2*5=10
The picture's dimensions will be 10"x15".
Given the CDF
![F_X(x)=\begin{cases}0&\text{for }x](https://tex.z-dn.net/?f=F_X%28x%29%3D%5Cbegin%7Bcases%7D0%26%5Ctext%7Bfor%20%7Dx%3C0%5C%5C1-e%5E%7B-8x%7D%26%5Ctext%7Bfor%20%7Dx%5Cge0%5Cend%7Bcases%7D)
we have PDF
![\dfrac{\mathrm dF_X(x)}{\mathrm dx}=f_X(x)=\begin{cases}0&\text{for }x](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20dF_X%28x%29%7D%7B%5Cmathrm%20dx%7D%3Df_X%28x%29%3D%5Cbegin%7Bcases%7D0%26%5Ctext%7Bfor%20%7Dx%3C0%5C%5C8e%5E%7B-8x%7D%26%5Ctext%7Bfor%20%7Dx%5Cge0%5Cend%7Bcases%7D)
Note that
![X](https://tex.z-dn.net/?f=X)
is wait-time given in hours, so we need to convert from minutes to hours:
![12\text{ min}\times\dfrac{1\text{ hr}}{60\text{ min}}=\dfrac15\text{ hr}](https://tex.z-dn.net/?f=12%5Ctext%7B%20min%7D%5Ctimes%5Cdfrac%7B1%5Ctext%7B%20hr%7D%7D%7B60%5Ctext%7B%20min%7D%7D%3D%5Cdfrac15%5Ctext%7B%20hr%7D)
so we're looking for
![\mathbb P\left(X](https://tex.z-dn.net/?f=%5Cmathbb%20P%5Cleft%28X%3C%5Cdfrac15%5Cright%29)
.
The CDF gives us this value right away, since
![F_X(x)=\mathbb P(X](https://tex.z-dn.net/?f=F_X%28x%29%3D%5Cmathbb%20P%28X%3Cx%29%3D%5Cmathbb%20P%28X%5Cle%20x%29)
for any continuous random variable
![X](https://tex.z-dn.net/?f=X)
with distribution function
![F_X(x)](https://tex.z-dn.net/?f=F_X%28x%29)
:
![\mathbb P\left(X](https://tex.z-dn.net/?f=%5Cmathbb%20P%5Cleft%28X%3C%5Cdfrac15%5Cright%29%3DF_X%5Cleft%28%5Cdfrac15%5Cright%29%3D1-e%5E%7B-8%2F5%7D%5Capprox0.7981)
To use the PDF, we need to integrate:
Answer:
1st option
Step-by-step explanation:
To find (f ○ g)(x) substitute g(x) into f(x) , that is
f(
)
= ![\frac{2}{\frac{1}{2x}+3 }](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B%5Cfrac%7B1%7D%7B2x%7D%2B3%20%7D)
= ![\frac{2}{\frac{1+6x}{2x} }](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B%5Cfrac%7B1%2B6x%7D%7B2x%7D%20%7D)
= ![\frac{4x}{1+6x}](https://tex.z-dn.net/?f=%5Cfrac%7B4x%7D%7B1%2B6x%7D)
Y2 - y1/x2 - x1 = slope
4 - 8/ 9 - 5 = slope
-4/4 = slope
-1 = slope
hope this helps :)