Question

1.

Answer 1

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)
$\frac{1}{x-1}+\frac{2}{x}=\frac{x}{x-1}$
We will first multiply each term by (x-1) to cancel those denominators:
$\frac{1}{x-1} \times \frac{x-1}{1}+\frac{2}{x} \times \frac{x-1}{1}=\frac{x}{x-1} \times \frac{x-1}{1} \\ \\ \frac{x-1}{x-1}+\frac{2(x-1)}{x}=\frac{x(x-1)}{x-1} \\ \\ 1+\frac{2(x-1)}{x}=x$

Now we multiply everything by x to cancel that denominator:
$1(x)+\frac{2(x-1)(x)}{x}=x(x) \\ \\x+2(x-1)=x^2 \\ \\x+2*x-2*1=x^2 \\x+2x-2=x^2 \\3x-2=x^2$

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.

Answer 2

Answer:

a

Step-by-step explanation: