Graph the solution set of each system of inequalities.
$x\le4$
$x\ge0$$y\ge0$
$x+2y\ge2$

Verified step by step guidance

1. Start by identifying the region for the inequality \( x \leq 4 \). This represents all points to the left of the vertical line \( x = 4 \) on the coordinate plane, including the line itself.
2. Next, consider the inequality \( x \geq 0 \). This represents all points to the right of the vertical line \( x = 0 \), including the line itself. The solution for \( x \) is the intersection of \( x \leq 4 \) and \( x \geq 0 \), which is the region between and including the lines \( x = 0 \) and \( x = 4 \).
3. Now, examine the inequality \( y \geq 0 \). This represents all points above the horizontal line \( y = 0 \), including the line itself. This means we are considering the upper half of the coordinate plane.
4. For the inequality \( x + 2y \geq 2 \), first rewrite it in slope-intercept form: \( y \geq -\frac{1}{2}x + 1 \). This represents the region above the line \( y = -\frac{1}{2}x + 1 \).
5. Finally, graph the solution set by shading the region that satisfies all the inequalities: \( x \leq 4 \), \( x \geq 0 \), \( y \geq 0 \), and \( y \geq -\frac{1}{2}x + 1 \). The solution set is the intersection of these regions on the coordinate plane.