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We begin by examining the lines as boundary lines to the shaded region. We see that the edge of the area consists of a vertical line along with the $y$-axis, a horizontal line along the $x$-axis, and a dashed line. Let's extend these lines.
The horizontal line can be described with the equation $y=0$ and the vertical line is given by $x=0.$ To find the equation of the dashed line we can mark the slope and the $y$-intercept. Then, the equation can be written in slope-intercept form.
Now we have the equations of the three boundary lines representing the region. The next step is to write inequalities for each equation to form a system of inequalities. We see that the shaded area is below, but not on, the line $y=\text{-} x+4,$ which means that the first inequality is $y < \text{-} x+4.$ The area is also located to the right of $x=0.$ This means it's all $x$-values greater than or equal $0.$ $x \geq 0.$ Finally, the area also above, and on, the line $y=0.$ Together these three inequalities describes the shaded region. $\begin{cases}y < \text{-} x+4 \\ x \geq 0 \\ y \geq0 \end{cases}$