1. The problem asks us to evaluate the definite integral $$\int_0^5 |2x - 5| \, dx$$. 2. To handle the absolute value, find where the expression inside changes sign: solve $$2x - 5 = 0$$. 3. Solving gives $$x = \frac{5}{2} = 2.5$$. 4. Split the integral at $$x=2.5$$: $$\int_0^5 |2x - 5| \, dx = \int_0^{2.5} |2x - 5| \, dx + \int_{2.5}^5 |2x - 5| \, dx$$. 5. For $$0 \leq x < 2.5$$, $$2x - 5 < 0$$, so $$|2x - 5| = -(2x - 5) = 5 - 2x$$. 6. For $$2.5 \leq x \leq 5$$, $$2x - 5 \geq 0$$, so $$|2x - 5| = 2x - 5$$. 7. Evaluate each integral: $$\int_0^{2.5} (5 - 2x) \, dx = \left[5x - x^2\right]_0^{2.5} = (5 \times 2.5 - (2.5)^2) - 0 = 12.5 - 6.25 = 6.25$$ $$\int_{2.5}^5 (2x - 5) \, dx = \left[x^2 - 5x\right]_{2.5}^5 = (25 - 25) - (6.25 - 12.5) = 0 - (-6.25) = 6.25$$ 8. Add the two results: $$6.25 + 6.25 = 12.5$$ 9. Express as a fraction: $$12.5 = \frac{25}{2}$$ Final answer: a. $$\frac{25}{2}$$