Solve (5x-1)(4x-1/4): Finding Values Where Function is Negative

To solve the problem of determining for which values of $x$ the function $y = (5x-1)(4x-\frac{1}{4})$ is negative, we will follow these steps:

• Step 1: Determine the roots of the function by setting each factor equal to zero.
• Step 2: Analyze the sign of the function on intervals defined by these roots.
• Step 3: Identify where the function is negative based on this analysis.

Let's proceed with these steps:

Step 1: Find the roots of the function.

To find the roots, set each factor equal to zero:

$5x - 1 = 0 \Rightarrow x = \frac{1}{5}$

$4x - \frac{1}{4} = 0 \Rightarrow x = \frac{1}{16}$

Step 2: Determine the intervals on the number line.

The roots divide the number line into the following intervals: $(-\infty, \frac{1}{16})$, $(\frac{1}{16}, \frac{1}{5})$, and $(\frac{1}{5}, \infty)$.

Step 3: Analyze the sign of the function in each interval:

• For $x < \frac{1}{16}$: Both $5x - 1$ and $4x - \frac{1}{4}$ are negative. Hence, their product is positive.
• For $\frac{1}{16} < x < \frac{1}{5}$: Here, $5x - 1$ is negative, and $4x - \frac{1}{4}$ is positive, making the product negative.
• For $x > \frac{1}{5}$: Both $5x - 1$ and $4x - \frac{1}{4}$ are positive, resulting in a positive product.

Now, consolidate the findings:

The function $y = (5x-1)(4x-\frac{1}{4})$ is less than zero for values $\frac{1}{16} < x < \frac{1}{5}$.

Therefore, the solution to the given problem is $\frac{1}{16} < x < \frac{1}{5}$.