Question 116812
solve and check each of the following equations for x.
x over 6 - x over 8 = 1
:
{{{x/6}}} - {{{x/8}}} = 1
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Find a value that is a multiple of 6 & 8. 24 is good one (you can use 48 also)
Multiply equation by 24
24*{{{x/6}}} - 24*{{{x/8}}} = 24(1)
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Cancel out the denominators and you have:
4x - 3x = 24
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x = 24
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Check solution in original equation:
{{{24/6}}} - {{{24/8}}} = 1
4 - 3 = 1
:
:
x over x - 2 MINUS x + 1 over x EQUALS 8 over x squared - 2x
{{{x/((x-2))}}} - {{{((x+1))/x}}} = {{{8/((x^2 - 2x))}}}
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You can factor the last denominator to
{{{x/((x-2))}}} - {{{((x+1))/x}}} = {{{8/(x(x - 2))}}}
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It's now apparent that x(x-2) is the common denominator of these fractions
Multiply equation by x(x-2)
x(x-2)*{{{x/((x-2))}}} - x(x-2)*{{{((x+1))/x}}} = x(x-2)*{{{8/(x(x - 2))}}}
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Cancel out the denominators and you have:
x*x - (x-2)(x+1) = 8
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x^2 - (x^2 - x - 2) = 8; FOILed (x-2)(x+1)
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Remove the brackets, change the signs
x^2 - x^2 + x + 2 = 8
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x^2's are eliminated, subtract 2 from both sides
x = 8 - 2
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x = 6
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Check solution in original equation:
{{{6/((6-2))}}} - {{{((6+1))/6}}} = {{{8/((6^2 - 2(6)))}}}
:
{{{6/4}}} - {{{7/6}}} = {{{8/24}}}
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Convert to a common denominator
{{{36/24}}} - {{{28/24}}} = {{{8/24}}}; confirms our solution of x=6
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Did this help you understand this stuff??