Solve: When is (5x-1)(4x-1/4) Greater Than Zero?

To solve this problem, we'll perform the following steps:

• Step 1: Identify the zeros of each factor.
• Step 2: Determine the sign of each factor across different intervals on the number line.
• Step 3: Identify where both factors give a positive product.

Now, let us work through each step:

Step 1: Find the values of $x$ where each factor equals zero:

• $5x - 1 = 0$ gives us $x = \frac{1}{5}$.
• $4x - \frac{1}{4} = 0$ gives us $x = \frac{1}{16}$.

These zeros divide the number line into intervals: $x < \frac{1}{16}$, $\frac{1}{16} < x < \frac{1}{5}$, and $x > \frac{1}{5}$.

Step 2: Analyze the sign of each factor in each interval:

• For $x < \frac{1}{16}$:
• $5x - 1$ is negative,
• $4x - \frac{1}{4}$ is negative,
• The product $(5x - 1)(4x - \frac{1}{4})$ is positive.
• For $\frac{1}{16} < x < \frac{1}{5}$:
• $5x - 1$ is negative,
• $4x - \frac{1}{4}$ is positive,
• The product $(5x - 1)(4x - \frac{1}{4})$ is negative.
• For $x > \frac{1}{5}$:
• $5x - 1$ is positive,
• $4x - \frac{1}{4}$ is positive,
• The product $(5x - 1)(4x - \frac{1}{4})$ is positive.

Step 3: Identify intervals where product is positive:

• $x < \frac{1}{16}$ and $x > \frac{1}{5}$.

Therefore, the solution to the inequality $y > 0$ is:
$x > \frac{1}{5}$ or $x < \frac{1}{16}$.