Questionolve for all values of x. $x-\frac{2}{x-8}=7$
Answer Attempt 1 out of 3

Studdy Solution

STEP 1

1. The equation $x - \frac{2}{x-8} = 7$ involves a rational expression.
2. We need to solve for $x$ by eliminating the fraction.

STEP 2

1. Eliminate the fraction by finding a common denominator.
2. Simplify the equation to a standard form.
3. Solve the resulting equation for $x$ .
4. Verify the solution(s) to ensure they do not make the denominator zero.

STEP 3

First, eliminate the fraction by multiplying every term by the denominator $x - 8$ to clear the fraction:
$x - \frac{2}{x-8} = 7$
Multiply through by $x - 8$ :
$(x - 8) \left( x - \frac{2}{x-8} \right) = 7(x - 8)$
This simplifies to:
$x(x - 8) - 2 = 7(x - 8)$

STEP 4

Expand and simplify both sides of the equation:
$x^2 - 8x - 2 = 7x - 56$
Rearrange all terms to one side to form a quadratic equation:
$x^2 - 8x - 2 - 7x + 56 = 0$
Combine like terms:
$x^2 - 15x + 54 = 0$

STEP 5

Solve the quadratic equation $x^2 - 15x + 54 = 0$ using the quadratic formula, factoring, or completing the square. Here, we will factor:
The quadratic factors as:
$(x - 9)(x - 6) = 0$
Set each factor equal to zero:
$x - 9 = 0 \quad \text{or} \quad x - 6 = 0$
Solve for $x$ :
$x = 9 \quad \text{or} \quad x = 6$

STEP 6

Verify the solutions to ensure they do not make the original denominator zero:
For $x = 9$ , the denominator $x - 8 = 1 \neq 0$ .
For $x = 6$ , the denominator $x - 8 = -2 \neq 0$ .
Both solutions are valid.
The values of $x$ are:
$\boxed{9} \quad \text{and} \quad \boxed{6}$

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