Question 79700
Given:
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{{{1/(x-3) +  1/(x+3) = 10/(x^2-9)}}}
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Don't think you are correct, but let's check your answer.  You got:
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{{{(x+3)+(x+3) = 10}}}
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Remove the parentheses and this becomes:
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{{{x + 3  + x + 3 = 10}}}
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Add like terms on the left side and you get:
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{{{2x + 6 = 10}}}
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Subtract 6 from both sides and you get:
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{{{2x = 4}}}
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Then divide both sides by 2 and you finally have {{{x = 2}}}.  If you return to the
original equation and substitute 2 for x it becomes:
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{{{1/(2-3)+1/(2+3) = 10/(2^2 -9)}}}
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Doing the math in the 3 denominators results in:
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{{{1/(-1) + 1/5 = 10/(4-9)}}}
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and this further simplifies to:
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{{{-1 +  1/5 = 10/-5}}}
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which becomes:
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{{{-1 + 1/5 = -2}}}
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by adding +1 to both sides this becomes:
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{{{+1/5 = -1}}}
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That doesn't look too good.
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Let's go back to the original problem and work it out:
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{{{1/(x-3) +  1/(x+3) = 10/(x^2-9)}}}
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You probably recognized that the denominator on the right side factors to:
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{{{(x-3)*(x+3)}}}
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Substitute that into the equation and it becomes:
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{{{1/(x-3) +  1/(x+3) = 10/((x-3)(x+3))}}}
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Now multiply all the terms on both sides by {{{(x-3)(x+3)}}}.  When you do it becomes:
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{{{((x-3)(x+3)*1)/(x-3) + ((x-3)(x+3)*1)/(x+3) = ((x-3)(x+3)*10)/((x-3)(x+3))}}}
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cross out the common terms in the denominator and the numerator:
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{{{(cross(x-3)(x+3)*1)/cross(x-3) + ((x-3)cross(x+3)*1)/cross(x+3) = (cross(x-3)cross(x+3)*10)/(cross(x-3)cross(x+3))}}}
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and the equation becomes:
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{{{(x+3) + (x-3) = 10}}}
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Remove the parentheses:
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{{{x + 3 + x - 3 = 10}}}
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Combine the like terms on the left side (note +3 -3 = 0) and you get:
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{{{2x = 10}}}
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Finally divide both sides by 2 and you get:
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{{{x = 5}}}
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Now let's check the answer.  Return to the original equation and substitute 5 for x to get:
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{{{1/(x-3) +  1/(x+3) = 10/(x^2-9)}}}
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{{{1/(5-3) +  1/(5+3) = 10/(5^2-9)}}}
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{{{1/2 + 1/8 = 10/(25-9)}}}
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{{{4/8 + 1/8 = 10/16}}}
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{{{5/8 = 10/16}}}
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{{{5/8 = 5/8}}}
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That works ... so the answer is x = 5
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Hope this helps you to find your mistake.