Answer
$$\frac{5}{2}$$
Work Step by Step
$$\eqalign{
& \int_0^2 {\left| {2x - 3} \right|dx} \cr
& {\text{By the definition of the aboslute value}} \cr
& \int_0^2 {\left| {2x - 3} \right|dx} = \int_0^{3/2} {\left[ { - \left( {2x - 3} \right)} \right]dx} + \int_{3/2}^2 {\left( {2x - 3} \right)dx} \cr
& {\text{ }} = - \int_0^{3/2} {\left( {2x - 3} \right)dx} + \int_{3/2}^2 {\left( {2x - 3} \right)dx} \cr
& {\text{Integrating}} \cr
& {\text{ }} = - \left[ {{x^2} - 3x} \right]_0^{3/2} + \left[ {{x^2} - 3x} \right]_{3/2}^2 \cr
& {\text{ }} = - \left[ {{{\left( {\frac{3}{2}} \right)}^2} - 3\left( {\frac{3}{2}} \right)} \right] + \left[ {{{\left( 2 \right)}^2} - 3\left( 2 \right)} \right] - \left[ {{{\left( 2 \right)}^2} - 3\left( 2 \right)} \right] \cr
& {\text{Simplify}} \cr
& {\text{ }} = \frac{9}{4} - 2 + \frac{9}{4} \cr
& {\text{ }} = \frac{5}{2} \cr} $$
