If sqrt(9x - 1) = sqrt(4x), then x =

1. Translate the problem requirements: We have an equation with square roots on both sides: \(\sqrt{9\mathrm{x} - 1} = \sqrt{4\mathrm{x}}\). We need to find the value of x that makes this equation true.
2. Apply the equality principle for square roots: Since both sides have square roots and they're equal, the expressions inside the square roots must also be equal.
3. Set up and solve the linear equation: Create the equation \(9\mathrm{x} - 1 = 4\mathrm{x}\) and solve for x using basic algebraic operations.
4. Verify the solution: Check that our answer satisfies the original equation and that both square root expressions are valid (non-negative).

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have in plain English. We're given an equation where the square root of one expression equals the square root of another expression: \(\sqrt{9\mathrm{x} - 1} = \sqrt{4\mathrm{x}}\).

Think of it this way: if I have two identical boxes, and each box contains the square root of something, then whatever is inside those boxes must be the same thing. This is our key insight.

We need to find the value of x that makes this statement true - that is, the value that makes both square root expressions equal to each other.

Process Skill: TRANSLATE - Converting the square root equation into a clear mathematical requirement

2. Apply the equality principle for square roots

Here's the beautiful simplicity of this problem: when two square roots are equal, the expressions under the square root signs must also be equal.

Think about it intuitively - if \(\sqrt{\mathrm{A}} = \sqrt{\mathrm{B}}\), then A must equal B. This is because the square root function gives us a unique positive result for each positive input.

So from \(\sqrt{9\mathrm{x} - 1} = \sqrt{4\mathrm{x}}\), we can conclude that:
\(9\mathrm{x} - 1 = 4\mathrm{x}\)

This transforms our square root equation into a simple linear equation that we can easily solve.

3. Set up and solve the linear equation

Now we have the linear equation: \(9\mathrm{x} - 1 = 4\mathrm{x}\)

Let's solve this step by step using basic algebra:
• Start with: \(9\mathrm{x} - 1 = 4\mathrm{x}\)
• Subtract 4x from both sides: \(9\mathrm{x} - 4\mathrm{x} - 1 = 4\mathrm{x} - 4\mathrm{x}\)
• Simplify: \(5\mathrm{x} - 1 = 0\)
• Add 1 to both sides: \(5\mathrm{x} = 1\)
• Divide both sides by 5: \(\mathrm{x} = \frac{1}{5}\)

So our answer is \(\mathrm{x} = \frac{1}{5}\).

4. Verify the solution

Let's check our answer by substituting \(\mathrm{x} = \frac{1}{5}\) back into the original equation.

Left side: \(\sqrt{9\mathrm{x} - 1} = \sqrt{9 \times \frac{1}{5} - 1} = \sqrt{\frac{9}{5} - 1} = \sqrt{\frac{9}{5} - \frac{5}{5}} = \sqrt{\frac{4}{5}}\)

Right side: \(\sqrt{4\mathrm{x}} = \sqrt{4 \times \frac{1}{5}} = \sqrt{\frac{4}{5}}\)

Perfect! Both sides equal \(\sqrt{\frac{4}{5}}\), which confirms our solution is correct.

We should also verify that both expressions under the square roots are non-negative (since we can't take square roots of negative numbers in the real number system):
• \(9\mathrm{x} - 1 = 9\left(\frac{1}{5}\right) - 1 = \frac{9}{5} - 1 = \frac{4}{5} > 0\) ✓
• \(4\mathrm{x} = 4\left(\frac{1}{5}\right) = \frac{4}{5} > 0\) ✓

Final Answer

The answer is \(\mathrm{x} = \frac{1}{5}\), which corresponds to choice D.

Our solution demonstrates the power of recognizing that equal square roots must have equal expressions underneath - this insight immediately simplified our problem from a square root equation to a basic linear equation.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the domain restrictions: Students often forget that square root expressions require their arguments to be non-negative. They may not realize they need to consider that both \(9\mathrm{x} - 1 \geq 0\) and \(4\mathrm{x} \geq 0\), which means \(\mathrm{x} \geq \frac{1}{9}\) and \(\mathrm{x} \geq 0\). While this doesn't affect the final answer in this case, ignoring domain restrictions can lead to extraneous solutions in more complex problems.

2. Attempting to solve by squaring both sides unnecessarily: Some students may immediately think to square both sides of \(\sqrt{9\mathrm{x} - 1} = \sqrt{4\mathrm{x}}\) to get \(9\mathrm{x} - 1 = 4\mathrm{x}\), without recognizing the more direct principle that if two square roots are equal, their arguments must be equal. This approach works but misses the elegant insight that makes the problem simpler.

Errors while executing the approach

1. Arithmetic errors in solving the linear equation: When solving \(9\mathrm{x} - 1 = 4\mathrm{x}\), students commonly make mistakes such as: incorrectly subtracting to get \(5\mathrm{x} + 1 = 0\) instead of \(5\mathrm{x} - 1 = 0\), or dividing incorrectly to get \(\mathrm{x} = 5\) instead of \(\mathrm{x} = \frac{1}{5}\). These basic algebraic errors are frequent under time pressure.

2. Sign errors when rearranging terms: Students may incorrectly handle the subtraction step, writing \(9\mathrm{x} - 4\mathrm{x} - 1 = 0\) as \(5\mathrm{x} + 1 = 0\) instead of \(5\mathrm{x} - 1 = 0\), leading to \(\mathrm{x} = -\frac{1}{5}\) instead of the correct \(\mathrm{x} = \frac{1}{5}\).

Errors while selecting the answer

1. Failing to verify the solution: Students may arrive at \(\mathrm{x} = \frac{1}{5}\) but not substitute back into the original equation to confirm their answer. This verification step would catch any computational errors and ensure both sides of the original equation are indeed equal.

2. Selecting the wrong fraction from the answer choices: After correctly calculating \(\mathrm{x} = \frac{1}{5}\), students might misread the answer choices and accidentally select \(\frac{1}{4}\) (choice C) or \(\frac{1}{7}\) (choice B) due to careless reading, especially since these fractions look similar at first glance.