In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. 3x+6y≤6, 2x+y≤8

Verified step by step guidance

1. Step 1: Identify each inequality in the system. Here, the system is \( \begin{cases} 2x + 4y \leq 8 \\ 4x + y \leq 4 \end{cases} \).
2. Step 2: Convert each inequality into an equation to find the boundary lines. For the first inequality, write \( 2x + 4y = 8 \), and for the second, write \( 4x + y = 4 \).
3. Step 3: Find the intercepts for each boundary line to help graph them. For \( 2x + 4y = 8 \), set \( x=0 \) to find \( y \)-intercept and set \( y=0 \) to find \( x \)-intercept. Repeat the same for \( 4x + y = 4 \).
4. Step 4: Graph the boundary lines on the coordinate plane using the intercepts. Since the inequalities are \( \leq \), shade the region below or on the line for each inequality.
5. Step 5: Determine the solution set by finding the overlapping shaded region that satisfies both inequalities simultaneously. This overlapping region represents the solution set of the system.

Key Concepts

Graphing Linear Inequalities

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

System of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the regions that satisfy each inequality individually. Graphing the system helps find the common shaded area representing all solutions that satisfy every inequality in the system.

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. This form makes it easier to plot the line accurately and determine which side to shade based on the inequality sign, facilitating the visualization of the solution region.