Solve the Rational Equations

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Exercise 1

\frac { 1 }{ { x }^{ 2 }-x} -\frac { 1 }{ x-1 } =0

Exercise 2

\frac { 1 }{ x-2 } +\frac { 1 }{ x+2 } =\frac { 1 }{ { x }^{ 2 }-4}

Exercise 3

\frac { 3 }{ x } =1+\frac { x-13 }{ 6 }

Exercise 4

x+\frac { 1 }{ x }=\frac { 26 }{ 5 }

Exercise 5

Two taps A and B fill a swimming pool together in two hours. Alone, it takes tap A three hours less than B to fill the same pool. How many hours does it take each tap to fill the pool separately?

Exercise 6

A faucet takes more than two hours longer to fill a tank than it would with a second faucet working at the same time as the first, where the job can be completed in 1 hour and 20 minutes. How long does it take to fill each one separately?

 

Solution of exercise 1

\frac { 1 }{ { x }^{ 2 }-x} -\frac { 1 }{ x-1 } =0

 

{ x }^{ 2 }-x=x(x-1)

LCM({ x }^{ 2 }-x,x-1)=x(x-1)

1-x=0

x=1

Verifying the solution:

\frac { 1 }{ 1-1 } -\frac { 1 }{ 1-1 } =0

\frac { 1 }{ 0 } -\frac { 1 }{ 0 } =0

The equation has no solution because for x = 1, the denominators are annulled.

 

Solution of exercise 2

\frac { 1 }{ x-2 } +\frac { 1 }{ x+2 } =\frac { 1 }{ { x }^{ 2 }-4}

{ x }^{ 2 }-4={ x }^{ 2 }-{ 2 }^{ 2 }=(x-2)(x+2)

L.C.M(x-2,x+2,{ x }^{ 2 }-4)=(x-2)(x+2)

 

x+2+x-2=1

2x=1

x=\frac { 1 }{ 2 }

 

Verifying the answer:

\frac { 1 }{ \frac { 1 }{ 2 }-2 } +\frac { 1 }{ \frac { 1 }{ 2 }+2 } =\frac { 1 }{ { \frac { 1 }{ 2 } }^{ 2 }-4}

\frac { 1 }{ \frac { -3 }{ 2 } } +\frac { 1 }{ \frac { 5 }{ 2 } } =\frac { 1 }{ \frac { -15 }{ 4 } }

\frac { -2 }{ 3 }+\frac { 2 }{ 5 }=\frac { -4 }{ 15 }

\frac { -4 }{ 15 }=\frac { -4 }{ 15 }

Solution of exercise 3

\frac { 3 }{ x } =1+\frac { x-13 }{ 6 }

L.C.M(x,6)=6x

18=6x+x(x-13)

18=6x+{ x }^{ 2 }-13x

{ x }^{ 2 }-7x-18=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-7)\pm \sqrt { { -7 }^{ 2 }-4(1)(-18) } }{ 2(1) }

x=\frac { 7\pm \sqrt { 49+72 } }{ 2 }

x=\frac { 7\pm \sqrt { 121 } }{ 2 }

x=\frac { 7\pm 11}{ 2 }

x=\frac { 7 + 11}{ 2 } \qquad x=\frac { 7 - 11}{ 2 }

x=\frac { 18 }{ 2 } \qquad x=\frac { -4 }{ 2 }

x=9 \qquad x=-2

 

Verifying the answer:

\frac { 3 }{ x } =1+\frac { x-13 }{ 6 }

For x=9:

\frac { 3 }{ 9 } =1+\frac { 9-13 }{ 6 }

\frac { 1 }{ 3 } =1+\frac { -4 }{ 6 }

\frac { 1 }{ 3 } =1+\frac { -2 }{ 3 }

\frac { 1 }{ 3 } =\frac { 3-2 }{ 3 }

\frac { 1 }{ 3 } =\frac { 1 }{ 3 }

For x=-2:

\frac { 3 }{ -2 } =1+(\frac { -2-13 }{ 6 })

-\frac { 3 }{ -2 } =1-\frac { 15 }{ 6 }

-\frac { 3 }{ 2 } =1-\frac { 5 }{ 2 }

-\frac { 3 }{ 2 } =\frac { 2-5 }{ 2 }

-\frac { 3 }{ 2 } =-\frac { 3 }{ 2 }

 

Solution of exercise 4

x+\frac { 1 }{ x }=\frac { 26 }{ 5 }

L.C.M(5,x)=5x

5{ x }^{ 2 }+5=26x

5{ x }^{ 2 }-26x+5=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-26)\pm \sqrt { { -26 }^{ 2 }-4(5)(5) } }{ 2(5) }

x=\frac { 26\pm \sqrt { 676-100 } }{ 10 }

x=\frac { 26\pm \sqrt { 576 } }{ 10 }

x=\frac { 26\pm 24 }{ 10 }

x=\frac { 26 + 24 }{ 10 } \qquad x=\frac { 26 - 24 }{ 10 }

x=\frac { 50 }{ 10 } \qquad x=\frac { 2 }{ 10 }

x=5 \qquad x=\frac { 1 }{ 5 }

 

Verifying the answer:

x+\frac { 1 }{ x }=\frac { 26 }{ 5 }

 

When x=5:

5+\frac { 1 }{ 5 }=\frac { 26 }{ 5 }

\frac { 25+1 }{ 5 }=\frac { 26 }{ 5 }

\frac { 26 }{ 5 }=\frac { 26 }{ 5 }

 

When x=\frac { 1 }{ 5 }:

\frac { 1 }{ 5 }+\frac { 1 }{ \frac { 1 }{ 5 } }=\frac { 26 }{ 5 }

\frac { 1 }{ 5 }+5\neq \frac { 26 }{ 5 }

 

Solution of exercise 5

Two taps A and B fill a swimming pool together in two hours. Alone, it takes tap A three hours less than B to fill the same pool. How many hours does it take each tap to fill the pool separately?

Time of A = x

 

Time of B = x + 3

\frac { 1 }{ A }+\frac { 1 }{ B }=\frac { 1 }{ 2 }

\frac { 1 }{ x }+\frac { 1 }{ x+3 }=\frac { 1 }{ 2 }

L.C.M(2,x,x+3)=2x(x+3)

2x+6+2x={ x }^{ 2 }+3x

{ x }^{ 2 }-x-6=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-1)\pm \sqrt { { (-1) }^{ 2 }-4(1)(-6) } }{ 2(1) }

x=\frac { 1\pm \sqrt { 1-24 } }{ 2 }

x=\frac { 1\pm \sqrt { 25 } }{ 2 }

x=\frac { 1\pm 5 }{ 2 }

x=\frac { 1 + 5 }{ 2 } \qquad x=\frac { 1 - 5 }{ 2 }

x=\frac { 6 }{ 2 } \qquad x=\frac { -4 }{ 2 }

x=3 \qquad x=-2

 

Verifying the answers:

\frac { 1 }{ x }+\frac { 1 }{ x+3 }=\frac { 1 }{ 2 }

 

When x=3:

\frac { 1 }{ 3 }+\frac { 1 }{ 3+3 }=\frac { 1 }{ 2 }

\frac { 1 }{ 3 }+\frac { 1 }{ 6 }=\frac { 1 }{ 2 }

\frac { 2+1 }{ 6 }=\frac { 1 }{ 2 }

\frac { 3 }{ 6 }=\frac { 1 }{ 2 }

\frac { 1 }{ 2 }=\frac { 1 }{ 2 }

 

When x=-2:

\frac { 1 }{ -2 }+\frac { 1 }{ -2+3 }=\frac { 1 }{ 2 }

-\frac { 1 }{ 2 }+\frac { 1 }{ 1 }=\frac { 1 }{ 2 }

-\frac { 1 }{ 2 }+1\neq\frac { 1 }{ 2 }

Time of A 3 hours

Time of B 6 hours

 

Solution of exercise 6

A faucet takes more than two hours longer to fill a tank than it would with a second faucet working at the same time as the first, where the job can be completed in 1 hour and 20 minutes. How long does it take to fill each one separately?

1st Time = x

2nd Time = x − 2

\frac { 1 }{ x }+\frac { 1 }{ x-2 }=\frac { 1 }{ \frac { 4 }{ 3 } }

3{ x }^{ 2 }-14x+8=0

x=\frac { -b\pm \sqrt { { b }^{ 2 }-4ac } }{ 2a }

x=\frac { -(-14)\pm \sqrt { { -14 }^{ 2 }-4(3)(8) } }{ 2(3) }

x=\frac { 14\pm \sqrt { 196-96 } }{ 6 }

x=\frac { 14\pm \sqrt { 100 } }{ 6 }

x=\frac { 14\pm 10}{ 6 }

x=\frac { 14 + 10}{ 6 }  \qquadx=\frac { 14 - 10}{ 6 } x=\frac { 24 }{ 6 }  \qquad x=\frac { 4 }{ 6 }

x=4  \qquad x=\frac { 2 }{ 3 }

1st Time 4 hours

2nd Time 2 hours

x=\frac { 2 }{ 3 } not a solution because the time for the second faucet would be negative.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.