Part A) x cannot be 1 or 0, because that would lead do division by 0. If x=1, 1-1=0. If x=0, then we have straight division by 0.
Part B) x=2 or x=1.
Together they could clean the house in 3.75 hours.
Explanation:
Part A is explained with the answer.
Part B)
[tex]\frac{1}{x-1}+\frac{2}{x}=\frac{x}{x-1}[/tex]
We will first multiply each term by (x-1) to cancel those denominators:
[tex]\frac{1}{x-1} \times \frac{x-1}{1}+\frac{2}{x} \times \frac{x-1}{1}=\frac{x}{x-1} \times \frac{x-1}{1}
\\
\\ \frac{x-1}{x-1}+\frac{2(x-1)}{x}=\frac{x(x-1)}{x-1}
\\
\\ 1+\frac{2(x-1)}{x}=x[/tex]
Now we multiply everything by x to cancel that denominator:
[tex]1(x)+\frac{2(x-1)(x)}{x}=x(x)
\\
\\x+2(x-1)=x^2
\\
\\x+2*x-2*1=x^2
\\x+2x-2=x^2
\\3x-2=x^2[/tex]
We want all of the variables to be on the same side of the equation, so subtract 3x from both sides:
3x-2-3x=x²-3x
-2=x²-3x
To solve quadratics, we set them equal to 0. Add 2 to both sides to make this happen:
-2+2=x²-3x+2
0=x²-3x+2
To solve this by factoring, we want factors of 2 that sum to -3. -2(-1)=2 and -2+-1=-3:
0=(x-2)(x-1)
The zero product property tells us that either x-2=0 or x-1=0; therefore x=2 or x=1.
Max cleans 1/10 of the house in an hour, since it takes him 10 hours to clean the whole house. Maggie cleans 1/6 of the house in an hour, since it takes her 6 hours to clean the whole house. Our equation is then:
1/10x+1/6x=1 (one whole, since it is one house; x is the number of hours)
Find a common denominator. 60 is the first thing that 10 and 6 both evenly divide into:
6/60x+10/60x=1
16/60x = 1
Divide both sides by 16/60:
16/60x ÷ 16/60 = 1 ÷ 16/60
x = 1/1 ÷ 16/60
x = 1/1 ×60/16
x = 60/16 = 3.75.
