Question

Solve the equation. Check for extraneous solutions. $$\sqrt{\frac{1}{9} x+1}-\frac{2}{3}=\frac{5}{3}$$

Short answer

The equation does not have any solution.

Step-by-step solution

Rearrange the equation to isolate the square root

Add \( \frac{2}{3} \) to both sides of the equation to isolate the square root part. This results in: \( \sqrt{\frac{1}{9} x+1}=1 \)

Square both sides of the equation

Squaring both sides eliminates the square root, resulting in the equation: \( \frac{1}{9} x+1 = 1 \)

Simplify the equation

Subtract 1 from both sides of the equation: \( \frac{1}{9} x = 0 \)

Multiply both sides by 9

Multiplying both sides by 9 isolates x on one side of the equation: \( x = 0 \)

Check for Extraneous Solutions

Substitute \( x = 0 \) into the original equation \( \sqrt{\frac{1}{9} \cdot 0+1} - \frac{2}{3} = \frac{5}{3} \). Simplifying gives \( \frac{2}{3} - \frac{2}{3} = \frac{5}{3} - \frac{2}{3} \), which simplifies to 0 = 1. Since this is not true, \( x = 0 \) is an extraneous solution of the equation.