Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}x \leq 2 \\y \geq -1\end{cases}$

Verified step by step guidance

1. Identify the inequalities in the system: \(x \leq 5\) and \(y \geq -6\).
2. Graph the boundary line for \(x \leq 5\). This is a vertical line at \(x = 5\). Since the inequality is \(\leq\), shade the region to the left of this line (including the line itself).
3. Graph the boundary line for \(y \geq -6\). This is a horizontal line at \(y = -6\). Since the inequality is \(\geq\), shade the region above this line (including the line itself).
4. The solution set to the system is the intersection of the two shaded regions: the area to the left of \(x = 5\) and above \(y = -6\).
5. Check a test point in the intersection region (for example, \((0,0)\)) to confirm it satisfies both inequalities.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves shading the region of the coordinate plane that satisfies the inequality. For example, x ≤ 5 means shading all points to the left of or on the vertical line x = 5. This visual representation helps identify all possible solutions.

Boundary Lines and Their Types

The boundary line of an inequality is the line where the inequality changes from true to false. For '≤' or '≥', the boundary line is solid, indicating points on the line satisfy the inequality. For '<' or '>', the line is dashed, meaning points on the line are not included.

Solution Set of a System of Inequalities

The solution set of a system of inequalities is the intersection of the shaded regions for each inequality. It includes all points that satisfy every inequality simultaneously. If no overlap exists, the system has no solution.