How do you solve #8/(x-4) -3=1/(x-10)#?

Answer 1

#x_(1,2) = (49 +- 7)/6#

The first thing to notice here is that you have two values of #x# for which the denominators are equal to zero.

This means that any possible solution set will not include thes values. In other words, you need

#x - 4 !=0 implies x != 4" "# and #" "x - 10 != 0 implies x != 10#
The next thing to do is use the common denominator of the two fractions, which is equal to #(x-4)(x-10)#, to rewrite the equation without denominators.
To do that, multiply the first fraction by #1 = (x-10)/(x-10)#, #3# by #1 = ((x-4)(x-10))/((x-4)(x-10))#, and the second fraction by #1 = (x-4)/(x-4)#.

This will get you

#8/(x-4) * (x-10)/(x-10) - 3 * ((x-4)(x-10))/((x-4)(x-10)) = 1/(x-10) * (x-4)/(x-4)#
#(8(x-10))/((x-4)(x-10)) - (3(x-4)(x-10))/((x-4)(x-10)) = (x-4)/((x-4)(x-10))#

This is of course equivalent to

#8x - 80 - 3(x^2 - 14x + 40) = x-4#
#7x - 76 - 3x^2 + 42x- 120 = 0#
#3x^2 - 49x +196 = 0#

Use the quadratic formula to find the two roots of this quadratic equation

#x_(1,2) = (-(-49) +- sqrt( (-49)^2 - 4 * 3 * 196))/(2 * 3)#
#x_(1,2) = (49 +- sqrt(49))/6 = (49 +- 7)/6#

Therefore, you have

#x_1 = (49 - 7)/6 = 7" "# and #x_2 = (49 + 7)/6 = 28/3#
Since both solutions satisfy the condtions #x !=4# and #x != 10#, both will be valid solutions to the original equation.
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Answer 2

To solve the equation ( \frac{8}{x-4} - 3 = \frac{1}{x-10} ), follow these steps:

  1. Find a common denominator for the fractions, which is ( (x-4)(x-10) ).
  2. Rewrite the equation with the common denominator: ( \frac{8(x-10)}{(x-4)(x-10)} - \frac{3(x-4)}{(x-4)(x-10)} = \frac{1}{x-10} ).
  3. Combine the fractions: ( \frac{8(x-10) - 3(x-4)}{(x-4)(x-10)} = \frac{1}{x-10} ).
  4. Distribute and simplify: ( \frac{8x - 80 - 3x + 12}{(x-4)(x-10)} = \frac{1}{x-10} ).
  5. Combine like terms: ( \frac{5x - 68}{(x-4)(x-10)} = \frac{1}{x-10} ).
  6. Cross multiply: ( (5x - 68)(x-10) = (x-4) ).
  7. Expand and simplify: ( 5x^2 - 50x - 68x + 680 = x - 4 ).
  8. Rearrange terms: ( 5x^2 - 118x + 684 = 0 ).
  9. Factor the quadratic equation, if possible, or use the quadratic formula to find the solutions for (x).
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Answer 3

To solve the equation ( \frac{8}{x-4} - 3 = \frac{1}{x-10} ):

  1. First, simplify the equation by finding a common denominator for the fractions.
  2. Then, combine like terms.
  3. Next, solve for ( x ).
  4. Finally, check for extraneous solutions.

The process involves cross-multiplication, simplification, and solving the resulting quadratic equation.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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