Question

Solve the equation. Check for extraneous solutions. $$\sqrt{\frac{1}{5} x-2}-\frac{1}{10}=\frac{7}{10}$$

Short answer

The solution to the equation is \(x = 13.2\). After checking, there are no extraneous solutions.

Step-by-step solution

Isolate the square root expression

To start off, the equation needs to be simplified by isolating the square root expression. This can be achieved by adding \(\frac{1}{10}\) to both sides of the equation. The equation is then: \[ \sqrt{\frac{1}{5} x-2} = \frac{8}{10} \] which simplifies further to: \[ \sqrt{\frac{1}{5} x-2} = \frac{4}{5} \]

Square both sides of the equation

To eliminate the square root, square both sides of the equation:\[ (\sqrt{\frac{1}{5}x - 2})^2 = (\frac{4}{5})^2 \] This simplifies to: \[ \frac{1}{5}x - 2 = \frac{16}{25} \]

Solve for x

Rearrange the equation to get x by itself. Addition of 2 to both sides yields: \[ \frac{1}{5}x = \frac{16}{25} + 2 \] which simplifies to: \[ \frac{1}{5}x = \frac{16}{25} + \frac{50}{25} \] further simplifies to: \[ \frac{1}{5}x = \frac{66}{25} \] and by multiplying both sides by 5, the solution to x is: \[ x = \frac{66}{5} = 13.2\]

Check for extraneous solutions

Substitute x = 13.2 into the original equation to check if it's a valid solution:\[ \sqrt{\frac{1}{5} * 13.2 - 2} - \frac{1}{10} = \frac{7}{10} \] which simplifies to \[ \frac{8}{10} - \frac{1}{10} = \frac{7}{10} \] Since both sides are equal, x = 13.2 is a valid solution and there are no extraneous solutions.