Problem 35 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The equation does not have any solution.

Step 1: 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: \( \sqrt{\frac{1}{9} x+1}=1 \)

Step 2: Square both sides of the equation

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

Step 3: Simplify the equation

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

Step 4: Multiply both sides by 9

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

Step 5: 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.

Key Concepts

Solving Radical Equations

Radical equations include variables under a square root, which can make them tricky to solve. The goal is to simplify and eventually remove the square root to find the value of the variable.

To start solving, you want to rearrange the equation. This often involves adding or subtracting terms to both sides. By doing this, you aim to have the square root alone on one side of the equation. Once that's done, you're ready to remove the square root by squaring both sides of the equation.

Important things to remember include:

• Maintain balance by performing the same operations on both sides.
• Proceed step by step, checking your work as you go.

Once the equation is simplified, you can isolate the variable and find potential solutions. But the process isn't finished there—you must verify your solutions due to possible extraneous solutions.

Isolating the Square Root

Isolating the square root is a crucial part of solving an equation with radicals. It means getting the square root by itself on one side of the equation.

In the exercise, this was accomplished by adding \( \frac{2}{3} \) to both sides. This left us with \( \sqrt{\frac{1}{9} x+1}=1 \).

Steps to isolate can include:

• Add or subtract terms from both sides to clear out other numbers.
• Multiply or divide if needed to simplify the coefficients of the square root.

After isolation, squaring both sides is the next step. This action removes the square root, giving you a linear equation to solve. This simplification allows you to focus solely on finding the variable's value.

Checking Solutions

Checking solutions is a must when dealing with radical equations because squaring both sides can introduce extraneous solutions.

An extraneous solution is a solution that appears from algebraic manipulation but doesn't satisfy the original equation.

After finding a potential solution, substitute it back into the original equation.
In our example, when we substituted \( x = 0 \) into the original equation, the equation became:

• \( \sqrt{\frac{1}{9} \cdot 0 + 1} - \frac{2}{3} = \frac{5}{3} \)
• \( \frac{2}{3} - \frac{2}{3} eq \frac{5}{3} - \frac{2}{3} \)

These calculations showed that the left side didn't equal the right side, indicating \( x = 0 \) is indeed extraneous. Confirm your solution aligns with the original equation to avoid these pitfalls.