In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x+2y≤4, y≥x−3

Verified step by step guidance

1. Step 1: Identify each inequality in the system. The first inequality is $x+5y\le 8$, and the second inequality is $y\ge x-6$.
2. Step 2: Graph the boundary lines for each inequality. For the first inequality, graph the line $x+5y=8$. For the second inequality, graph the line $y=x-6$. Use a solid line for both because the inequalities include equal to (≤ and ≥).
3. Step 3: Determine the shading region for each inequality. For 8, choose a test point not on the line (like (0,0)) and check if it satisfies the inequality. If it does, shade the side containing that point; if not, shade the opposite side. Repeat this for the second inequality .
4. Step 4: Identify the solution set as the region where the shaded areas of both inequalities overlap. This overlapping region represents all points (x,y) that satisfy both inequalities simultaneously.
5. Step 5: Label the graph clearly, including the boundary lines and shaded regions, to visually represent the solution set of the system of inequalities.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves shading the region of the coordinate plane that satisfies the inequality. The boundary line, given by the corresponding equation, is drawn solid if the inequality includes equality (≤ or ≥) and dashed if it does not (< or >). This visual representation helps identify all possible solutions.

System of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the individual solution regions, representing all points that satisfy every inequality in the system. Graphing helps visualize this common solution area.

Slope-Intercept Form and Boundary Lines

Converting inequalities to slope-intercept form (y = mx + b) simplifies graphing by identifying the slope and y-intercept of the boundary line. For example, y ≥ x - 6 has slope 1 and y-intercept -6, guiding the drawing of the boundary and shading above it for solutions.