Graph the solution set of each system of inequalities.
$x\le 4$
$x\ge 0$
$y\ge 0$
$x+2y\ge 2$

Verified step by step guidance

1. Identify each inequality and understand what it represents on the coordinate plane. For example, the inequality \(x \leq 4\) represents all points to the left of and including the vertical line \(x = 4\).
2. Graph the boundary lines for each inequality. For \(x \leq 4\), draw the vertical line \(x = 4\). For \(x \geq 0\), draw the vertical line \(x = 0\). For \(y \geq 0\), draw the horizontal line \(y = 0\). For \(x + 2y \geq 2\), first rewrite it as \$2y \geq 2 - x\( and then \)y \geq \frac{2 - x}{2}\(, which is a line with slope \)-\frac{1}{2}\( and y-intercept \)1$.
3. Determine which side of each boundary line to shade by testing a point not on the line (commonly the origin \((0,0)\) if it is not on the line). For example, for \(x \leq 4\), test \((0,0)\): since \$0 \leq 4\( is true, shade to the left of \)x=4$. Repeat this for each inequality.
4. Combine all shaded regions by finding the intersection where all inequalities are true simultaneously. This overlapping region is the solution set to the system of inequalities.
5. Label the final solution region clearly on the graph, ensuring the boundaries are solid lines (since inequalities include equality) and the correct region is shaded.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line represented by the related equation and then shading the region that satisfies the inequality. For example, the inequality x ≤ 4 is graphed by drawing a vertical line at x = 4 and shading all points to the left, including the line if the inequality is ≤ or ≥.

System of Inequalities

A system of inequalities consists of two or more inequalities considered together. The solution set is the intersection of the regions that satisfy each inequality individually. Graphing the system means finding the common shaded area that meets all conditions simultaneously.

Interpreting Inequalities in Two Variables

Inequalities involving two variables, like x + 2y ≥ 2, define half-planes on the coordinate plane. To graph them, rewrite the inequality as an equation to find the boundary line, then test points to determine which side satisfies the inequality, shading that region accordingly.