$\int_0^2 |4x - 5|\,dx = \ldots$

The expression inside the absolute value is zero when $4x - 5 = 0 \Rightarrow x = \dfrac{5}{4}$.

The interval of integration is from $0$ to $2$, so we split the integral at $x = \dfrac{5}{4}$.

$\int_{0}^{2} |4x - 5| dx = \int_{0}^{\dfrac{5}{4}} -(4x - 5) dx + \int_{\dfrac{5}{4}}^{2} (4x - 5) dx$

$= \int_{0}^{\dfrac{5}{4}} (5 - 4x) dx + \int_{\dfrac{5}{4}}^{2} (4x - 5) dx$

Evaluating the first integral:

$\int_{0}^{\dfrac{5}{4}} (5 - 4x) dx = \left[ 5x - 2x^2 \right]_{0}^{\dfrac{5}{4}}$

$= 5\left(\dfrac{5}{4}\right) - 2\left(\dfrac{5}{4}\right)^2$

$= \dfrac{25}{4} - 2\left(\dfrac{25}{16}\right) = \dfrac{25}{4} - \dfrac{25}{8} = \dfrac{50}{8} - \dfrac{25}{8} = \dfrac{25}{8}$

Evaluating the second integral:

$\int_{\dfrac{5}{4}}^{2} (4x - 5) dx = \left[ 2x^2 - 5x \right]_{\dfrac{5}{4}}^{2}$

$= \left( 2(2)^2 - 5(2) \right) - \left( 2\left(\dfrac{5}{4}\right)^2 - 5\left(\dfrac{5}{4}\right) \right)$

$= (8 - 10) - \left( 2\left(\dfrac{25}{16}\right) - \dfrac{25}{4} \right)$

$= -2 - \left( \dfrac{25}{8} - \dfrac{25}{4} \right) = -2 - \left( \dfrac{25 - 50}{8} \right)$

$= -2 - \left( -\dfrac{25}{8} \right) = -2 + \dfrac{25}{8} = \dfrac{-16 + 25}{8} = \dfrac{9}{8}$

Adding the two parts together:

$\int_{0}^{2} |4x - 5| dx = \dfrac{25}{8} + \dfrac{9}{8} = \dfrac{34}{8} = \dfrac{17}{4}$