Solve (2x-1/2)(x-2¼): Find When Function is Positive

Look at the function below:

$y=\left(2x-\frac{1}{2}\right)\left(x-2\frac{1}{4}\right)$

Then determine for which values of $x$ the following is true:

$f(x) > 0$

To solve this problem, we will examine the intervals defined by the roots of the quadratic function.

Step 1: Find the roots of each factor:
For $2x - \frac{1}{2} = 0$, solve for $x$:
$2x = \frac{1}{2} \quad \Rightarrow \quad x = \frac{1}{4}$
For $x - 2\frac{1}{4} = 0$, solve for $x$:
$x = 2\frac{1}{4}$

Step 2: Determine the test intervals around these roots, which are $x < \frac{1}{4}$, $\frac{1}{4} < x < 2\frac{1}{4}$, and $x > 2\frac{1}{4}$.

Step 3: Test each interval to determine where the product is positive:

• For $x < \frac{1}{4}$, both factors $(2x - \frac{1}{2})$ and $(x - 2\frac{1}{4})$ are negative, so the product is positive.
• For $\frac{1}{4} < x < 2\frac{1}{4}$, one factor is positive $(2x - \frac{1}{2})$ and the other is negative $(x - 2\frac{1}{4})$, resulting in a negative product.
• For $x > 2\frac{1}{4}$, both factors $(2x - \frac{1}{2})$ and $(x - 2\frac{1}{4})$ are positive, so the product is positive.

Therefore, the solution for $f(x) > 0$ is when $x > 2\frac{1}{4}$ or $x < \frac{1}{4}$.

The correct choice that matches this analysis is:
$x > 2\frac{1}{4}$ or $x < \frac{1}{4}$.