Problem 13 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{3 x-1}+1=4\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{10}{3}\).

Step 1: Isolate the Square Root

Start by isolating the square root expression. Subtract 1 from both sides of the equation: \[ \sqrt{3x - 1} + 1 = 4 \rightarrow \sqrt{3x - 1} = 3 \]

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \[ (\sqrt{3x - 1})^2 = 3^2 \] This simplifies to: \[ 3x - 1 = 9 \]

Step 3: Solve for x

Solve for \(x\) by first adding 1 to both sides and then dividing by 3: \[ 3x - 1 + 1 = 9 + 1 \rightarrow 3x = 10 \] \[ x = \frac{10}{3} \]

Step 4: Check the Solution

Substitute \(x = \frac{10}{3}\) back into the original equation to verify: \[ \sqrt{3\left(\frac{10}{3}\right) - 1} + 1 = 4 \] Simplify inside the square root: \[ \sqrt{10 - 1} = \sqrt{9} = 3 \] So the equation becomes: \[ 3 + 1 = 4 \] This checks out, confirming \(x = \frac{10}{3}\) is correct.

Key Concepts

Isolate the Square Root

When faced with a radical equation, the first step is to isolate the square root on one side of the equation. This makes it easier to solve the equation by removing the square root in the next steps. For example, in the equation \( \sqrt{3x - 1} + 1 = 4 \), we want to get the square root alone. We do this by subtracting 1 from both sides:

• This action leaves us with \( \sqrt{3x - 1} = 3 \).

Understanding the need to isolate the square root is crucial, as it sets up the equation to be manipulated in simpler ways. Without isolating the square root, additional terms could complicate the next steps.
Always remember:

• Whatever operation you perform on one side of the equation, you must also perform on the other.
• Isolating the square root often involves basic operations like addition, subtraction, division, or multiplication.

Squaring Both Sides

Once the square root is isolated, the next step is to remove it by squaring both sides of the equation. This action is powerful because squaring the square root cancels it out, leaving a simpler equation to solve. For example, if you have:

• \( \sqrt{3x - 1} = 3 \)
• Squaring both sides leads to \( (\sqrt{3x - 1})^2 = 3^2 \)

This simplifies to:

• \( 3x - 1 = 9 \)

Why do we square both sides? Because it transforms the radical equation, a potentially difficult type to solve, into a linear equation, which is generally much simpler. However, it's important to remember that squaring both sides of an equation can introduce extraneous solutions. That's why the final step in our process is crucial.

Checking Solutions

After solving the equation by isolating the square root and squaring both sides, it's essential to check your solutions. This step verifies the correctness of your answer and ensures no extraneous solutions have been introduced during the squaring process.
To check your solution, substitute the found value of \(x\) back into the original equation. For the equation \( \sqrt{3x - 1} + 1 = 4 \), if we find \(x = \frac{10}{3}\), plug it back in:

• Substitute to get \( \sqrt{3\left(\frac{10}{3}\right) - 1} + 1 \)
• This simplifies to \( \sqrt{10 - 1} + 1 = \sqrt{9} + 1 = 4 \)

This confirms our solution is correct because both sides of the original equation are equal after substitution.
Key reasons to perform this check are:

• Confirm the solution is valid.
• Ensure any errors during calculation are caught.
• Remove the risk of accepting an incorrect extraneous solution.