SOLUTION: solve for x: (8/x-1)+(7x/x+1)=7

Question 326167: solve for x:
(8/x-1)+(7x/x+1)=7

Answer by Apathious(24) (Show Source): You can put this solution on YOUR website!
((8)/(x-1)+((7x)/(x+1)))=7
Multiply each term by a factor of 1 that will equate all the denominators. In this case, all terms need a denominator of (x+1)(x-1). The (8)/((x-1)) expression needs to be multiplied by ((x+1))/((x+1)) to make the denominator (x+1)(x-1). The (7x)/((x+1)) expression needs to be multiplied by ((x-1))/((x-1)) to make the denominator (x+1)(x-1).
((8)/(x-1)*(x+1)/(x+1)+(7x)/(x+1)*(x-1)/(x-1))=7
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of (x+1)(x-1).
((8(x+1))/((x+1)(x-1))+(7x)/(x+1)*(x-1)/(x-1))=7
Multiply the expression by a factor of 1 to create the least common denominator (LCD) of (x+1)(x-1).
((8(x+1))/((x+1)(x-1))+(7x(x-1))/((x+1)(x-1)))=7
The numerators of expressions that have equal denominators can be combined. In this case, (8(x+1))/((x+1)(x-1)) and ((7x(x-1)))/((x+1)(x-1)) have the same denominator of (x+1)(x-1), so the numerators can be combined.
((8(x+1)+(7x(x-1)))/((x+1)(x-1)))=7
Simplify the numerator of the expression.
((8x+8+7x^(2)-7x)/((x+1)(x-1)))=7
Since 8x and -7x are like terms, add -7x to 8x to get x.
((x+8+7x^(2))/((x+1)(x-1)))=7
Reorder the polynomial x+8+7x^(2) alphabetically from left to right, starting with the highest order term.
((7x^(2)+x+8)/((x+1)(x-1)))=7
Remove the parentheses around the expression ((7x^(2)+x+8))/((x+1)(x-1)).
(7x^(2)+x+8)/((x+1)(x-1))=7
Since the variable is in the denominator on the left-hand side of the equation, this can be solved as a ratio. For example, (A)/(B)=C is equivalent to (A)/(C)=B.
(7x^(2)+x+8)/(7)=(x+1)(x-1)
Find the LCD (least common denominator) of ((7x^(2)+x+8))/(7)+(x+1)(x-1).
Least common denominator: 7
Multiply each term in the equation by 7 in order to remove all the denominators from the equation.
(7x^(2)+x+8)/(7)*7=(x+1)(x-1)*7
Simplify the left-hand side of the equation by canceling the common factors.
7x^(2)+x+8=(x+1)(x-1)*7
Simplify the right-hand side of the equation by multiplying out all the terms.
7x^(2)+x+8=7x^(2)-7
Since 7x^(2) contains the variable to solve for, move it to the left-hand side of the equation by subtracting 7x^(2) from both sides.
7x^(2)+x+8-7x^(2)=-7
Since 7x^(2) and -7x^(2) are like terms, add -7x^(2) to 7x^(2) to get 0.
0+x+8=-7
Combine all similar terms in the polynomial 7x^(2)+x+8-7x^(2).
x+8=-7
Since 8 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 8 from both sides.
x=-8-7
Subtract 7 from -8 to get -15.
x=-15
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