To solve the rational equation
[tex]\[
\frac{x+6}{x-3} = \frac{4}{7},
\][/tex]
we will follow a step-by-step approach.
1. Cross-multiply to eliminate the fractions:
Multiply both sides of the equation by [tex]\((x - 3)\)[/tex] and 7 to get rid of the fractions.
[tex]\[
7(x + 6) = 4(x - 3)
\][/tex]
2. Distribute and simplify:
Distribute the numbers 7 and 4 on both sides of the equation.
[tex]\[
7x + 42 = 4x - 12
\][/tex]
3. Isolate the variable [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], you need to get all [tex]\(x\)[/tex] terms on one side and constants on the other. Subtract [tex]\(4x\)[/tex] from both sides.
[tex]\[
7x - 4x + 42 = -12
\][/tex]
Simplify.
[tex]\[
3x + 42 = -12
\][/tex]
Now, subtract 42 from both sides.
[tex]\[
3x = -12 - 42
\][/tex]
Simplify.
[tex]\[
3x = -54
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 3.
[tex]\[
x = \frac{-54}{3}
\][/tex]
Simplify.
[tex]\[
x = -18
\][/tex]
Thus, the correct solution to the given rational equation is:
[tex]\[
\boxed{x = -18}
\][/tex]
Among the given choices, the correct one is:
D. [tex]\(x = -18\)[/tex]
