To solve the equation:
[tex]\[
-\frac{5}{2x - 12} + 1 = \frac{8}{x - 6}
\][/tex]
Let's go through the steps logically:
1. Set up the equation:
Given the equation,
[tex]\[
-\frac{5}{2x - 12} + 1 = \frac{8}{x - 6}
\][/tex]
2. Isolate the fractions:
Move [tex]\(1\)[/tex] to the right side by subtracting [tex]\(1\)[/tex] from both sides:
[tex]\[
-\frac{5}{2x - 12} = \frac{8}{x - 6} - 1
\][/tex]
3. Common denominator:
To combine the terms on the right side, they need a common denominator. Thus:
[tex]\[
\frac{8}{x - 6} - 1 = \frac{8}{x - 6} - \frac{x - 6}{x - 6} = \frac{8 - (x - 6)}{x - 6} = \frac{8 - x + 6}{x - 6} = \frac{14 - x}{x - 6}
\][/tex]
Therefore, we have:
[tex]\[
-\frac{5}{2x - 12} = \frac{14 - x}{x - 6}
\][/tex]
4. Observe the form of the denominators:
Notice that [tex]\(2x - 12 = 2(x - 6)\)[/tex].
Simplify by expressing [tex]\(2x - 12\)[/tex] as [tex]\(2(x - 6)\)[/tex]:
[tex]\[
-\frac{5}{2(x - 6)} = \frac{14 - x}{x - 6}
\][/tex]
5. Multiplying both sides by a common factor:
Multiply both sides by [tex]\(2(x - 6)\)[/tex] to clear the denominators:
[tex]\[
-5 = (14 - x) \cdot 2
\][/tex]
Simplifying inside the parentheses:
[tex]\[
-5 = 28 - 2x
\][/tex]
6. Solving for [tex]\(x\)[/tex]:
Add [tex]\(2x\)[/tex] to both sides:
[tex]\[
2x - 5 = 28
\][/tex]
Add [tex]\(5\)[/tex] to both sides:
[tex]\[
2x = 33
\][/tex]
Divide both sides by [tex]\(2\)[/tex]:
[tex]\[
x = \frac{33}{2}
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = \frac{33}{2}
\][/tex]
