Question

Directions - Solve the radical equation.



​
11
−
x
=
x
−
7
11−x
​
=
x−7
​

Answer 1

To solve the equation

\[
\frac{11 - x}{11} = x - 7,
\]

let's first eliminate the fraction by multiplying both sides of the equation by 11. This gives us:

\[
11 - x = 11(x - 7).
\]

Now, distribute 11 on the right side:

\[
11 - x = 11x - 77.
\]

Next, we want to get all terms involving \(x\) on one side and the constant terms on the other. We can do this by adding \(x\) and 77 to both sides:

\[
11 + 77 = 11x + x,
\]

which simplifies to:

\[
88 = 12x.
\]

Now, divide both sides by 12 to solve for \(x\):

\[
x = \frac{88}{12} = \frac{22}{3}.
\]

We can now check if \(x = \frac{22}{3}\) satisfies the original equation.

Substituting \(x = \frac{22}{3}\) back into the original equation:

Left side:
\[
\frac{11 - \frac{22}{3}}{11} = \frac{\frac{33}{3} - \frac{22}{3}}{11} = \frac{\frac{11}{3}}{11} = \frac{11}{33} = \frac{1}{3}.
\]

Right side:
\[
\frac{22}{3} - 7 = \frac{22}{3} - \frac{21}{3} = \frac{1}{3}.
\]

Since both sides of the equation are equal:

\[
\frac{1}{3} = \frac{1}{3},
\]

the solution is verified.

Thus, the solution to the equation is

\[
x = \frac{22}{3}.
\]