1. (05.08 mc) given the equation 1 over x−1 + 2 over x =x over x−1 part a: what values of x that would make the equation no solution? why? part b: what is the solution of the equation. show all of your steps. max and maggie have to clean the house. it takes max 10 hours to clean the house, while maggie can complete the task in 6 hours. how long will it take for them to work together? explain each step in solving this equation.

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)

We will first multiply each term by (x-1) to cancel those denominators:


Now we multiply everything by x to cancel that denominator:


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.