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rational equations











































If at least one of the terms of an equation is a fraction, the equation is a rational equation.

The equation shown below is a rational equation because two of the terms are fractions.

how do i solve this equation?

(x)/(x+3) + (1)/(x-3) = 1

Comments for rational equations

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Apr 07, 2011
Rational Equations
by: Staff


The question:

how do i solve this equation (x)/(x+3)+(1)/(x-3)=1


The answer:

(x)/(x + 3) + (1)/(x - 3) = 1

To add these two fractions, determine a common denominator:

Multiply the first fraction by another fraction: (x - 3)/(x - 3)

Note that (x - 3)/(x - 3) = 1, so you are only multiplying the first fraction by 1.


Multiply the second fraction by another fraction: (x + 3)/(x + 3)

Note that (x + 3)/(x + 3) = 1, so you are only multiplying the second fraction by 1.


[(x)/(x + 3)]*[(x - 3)/(x - 3)] + [(1)/(x - 3)]*[(x + 3)/(x + 3)] = 1


Multiply the numerators of the two fractions in the first term together, and multiply the denominators of the two fractions in the first term together.


Multiply the numerators of the two fractions in the second term together, and multiply the denominators of the two fractions in the second term together.



[(x) * (x - 3)]/[(x + 3) * (x - 3)] + [(1) * (x + 3)]/[(x - 3) * (x + 3)] = 1

(x² - 3x)/[(x + 3) * (x - 3)] + [(1) * (x + 3)]/[(x - 3) * (x + 3)] = 1

(x² - 3x)/( x² - 9) + [(1) * (x + 3)]/[(x - 3) * (x + 3)] = 1

(x² - 3x)/( x² - 9) + (x + 3)/( x² - 9) = 1

The denominators of both fractions are the same: ( x² - 9)

Add the two fractions:

(x² - 3x)/( x² - 9) + (x + 3)/( x² - 9) = 1

[(x² - 3x) + (x + 3)] /( x² - 9) = 1

Combine like terms

[(x² - 3x) + (x + 3)] /( x² - 9) = 1

(x² - 3x + x + 3) /( x² - 9) = 1

(x² - 2x + 3)/( x² - 9) = 1

Multiply each side of the equation by ( x² - 9)
[(x² - 2x + 3)/( x² - 9)] * ( x² - 9) = 1 * ( x² - 9)

(x² - 2x + 3) * [( x² - 9)/( x² - 9)] = 1 * ( x² - 9)

(x² - 2x + 3) * 1 = 1 * ( x² - 9)

(x² - 2x + 3) = 1 * ( x² - 9)

x² - 2x + 3 = 1 * ( x² - 9)

x² - 2x + 3 = ( x² - 9)

x² - 2x + 3 = x² - 9

subtract x² from each side of the equation

x² - 2x + 3 = x² - 9

x² - x² - 2x + 3 = x² - x² - 9

0 - 2x + 3 = 0 - 9

- 2x + 3 = - 9

Add 2x to each side of the equation

- 2x + 2x + 3 = - 9 + 2x

0 + 3 = - 9 + 2x

3 = - 9 + 2x

Add 9 to each side of the equation

3 + 9 = - 9 + 9 + 2x

3 + 9 = 0 + 2x

3 + 9 = 2x

12 = 2x

Divide each side of the equation by 2

12/2 = 2x/2

6 = 2x/2

6 = x*(2/2)

6 = x*1

6 = x

The final answer is: x = 6


Check the answer by substituting 6 for every x in the original equation:

(x)/(x + 3) + (1)/(x - 3) = 1

(6)/(6 + 3) + (1)/(6 - 3) = 1

(6)/(9) + (1)/(3) = 1

6/9 + 1/3 = 1

2/3 + 1/3 = 1

1 = 1, correct

Since 1 does = 1, the solution of x = 6 is correct.




Thanks for writing.


Staff
www.solving-math-problems.com


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