In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x≤2, y≥−1

Verified step by step guidance

1. Step 1: Understand the system of inequalities. The system consists of two inequalities: $x\le 5$ and $y\ge -6$. This means we are looking for all points (x, y) where x is less than or equal to 5 and y is greater than or equal to -6.
2. Step 2: Graph the boundary lines for each inequality. For $x\le 5$, draw a vertical line at $x=5$. For $y\ge -6$, draw a horizontal line at $y=-6$. These lines divide the plane into regions.
3. Step 3: Determine the shading for each inequality. Since $x\le 5$, shade the region to the left of the vertical line (including the line). For $y\ge -6$, shade the region above the horizontal line (including the line).
4. Step 4: Identify the solution set as the intersection of the shaded regions. The solution set consists of all points that satisfy both inequalities simultaneously, so it is the area where the shaded regions overlap.
5. Step 5: Verify the solution set by testing a point in the overlapping region, such as (0, 0). Check if it satisfies both inequalities: $0\le 5$ and $0\ge -6$. Since both are true, the shading is correct.

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.