Let's analyze the equation $f(x) = g(x)$, where:

• $f(x) = |2x - 1|$
• $g(x) = \frac{x}{4}$

We are tasked with solving the equation:

$|2x - 1| = \frac{x}{4}$

Step 1: Solve for $x$

The absolute value equation $|2x - 1| = \frac{x}{4}$ can be rewritten as two separate cases:

Case 1: $2x - 1 = \frac{x}{4}$ Multiply both sides by 4 to eliminate the denominator: $4(2x - 1) = x$ $8x - 4 = x$ Now, solve for $x$: $8x - x = 4$ $7x = 4$ $x = \frac{4}{7}$

Case 2: $-(2x - 1) = \frac{x}{4}$ Multiply both sides by 4: $4(-(2x - 1)) = x$ $-8x + 4 = x$ Now, solve for $x$: $-8x - x = -4$ $-9x = -4$ $x = \frac{4}{9}$

Step 2: Verify the Solutions

For $x = \frac{4}{7}$:

• $f(x) = |2(\frac{4}{7}) - 1| = | \frac{8}{7} - 1| = \frac{1}{7}$
• $g(x) = \frac{\frac{4}{7}}{4} = \frac{4}{28} = \frac{1}{7}$

Both sides are equal, so $x = \frac{4}{7}$ is a solution.

For $x = \frac{4}{9}$:

• $f(x) = |2(\frac{4}{9}) - 1| = | \frac{8}{9} - 1| = \frac{1}{9}$
• $g(x) = \frac{\frac{4}{9}}{4} = \frac{4}{36} = \frac{1}{9}$

Both sides are equal, so $x = \frac{4}{9}$ is also a solution.

Conclusion

The solutions to the equation are $x = \frac{4}{7}$ and $x = \frac{4}{9}$.

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Would you like more details about any of the steps or need further clarification?

Here are some related questions for you:

1. How would you solve an absolute value equation involving a linear function on one side and a quadratic on the other?
2. What does it mean for a function to have absolute value, and how does it affect the solutions?
3. How can we graph $|2x - 1|$ and $\frac{x}{4}$ to visually interpret the solutions?
4. What are the different types of equations that can be solved using absolute value functions?
5. How would you solve for intersections of other types of functions (e.g., polynomial vs. rational functions)?

Tip: Always check for extraneous solutions when solving equations with absolute values, as some cases may produce values that don't satisfy the original equation.