Question: t
1
2 + \frac{1}{1 - \frac{1}{x}} = \frac{2}{5}, find the value of x.
1. **State the problem:**
Solve for $x$ in the equation:
$$2 + \frac{1}{1 - \frac{1}{x}} = \frac{2}{5}$$
2. **Rewrite the equation and understand the structure:**
The equation has a complex fraction in the denominator:
$$2 + \frac{1}{1 - \frac{1}{x}} = \frac{2}{5}$$
3. **Isolate the fraction term:**
Subtract 2 from both sides:
$$\frac{1}{1 - \frac{1}{x}} = \frac{2}{5} - 2$$
Calculate the right side:
$$\frac{2}{5} - 2 = \frac{2}{5} - \frac{10}{5} = -\frac{8}{5}$$
So:
$$\frac{1}{1 - \frac{1}{x}} = -\frac{8}{5}$$
4. **Invert both sides to solve for the denominator:**
$$1 - \frac{1}{x} = \frac{1}{-\frac{8}{5}} = -\frac{5}{8}$$
5. **Solve for $\frac{1}{x}$:**
$$1 - \frac{1}{x} = -\frac{5}{8}$$
Subtract 1 from both sides:
$$- \frac{1}{x} = -\frac{5}{8} - 1 = -\frac{5}{8} - \frac{8}{8} = -\frac{13}{8}$$
Multiply both sides by $-1$:
$$\frac{1}{x} = \frac{13}{8}$$
6. **Solve for $x$:**
$$x = \frac{8}{13}$$
**Final answer:**
$$x = \frac{8}{13}$$