Graph the solution set of each system of inequalities or indicate that the system has no solution. { 3 x + 6 y ≤ 6 2 x + y ≤ 8 \begin{cases}3x + 6y \leq 6 \\2x + y \leq 8\end{cases} { 3 x + 6 y ≤ 6 2 x + y ≤ 8

Consider the following system of inequalities two X plus four, Y is less than equal to eight and four, X plus Y is less than equal to four. Now, I want to graph the solution set so this will be the shaded region that qualifies both of our inequalities. We look here, our first line to expose for Y is less than equals. Eight is denoted by the green line, which is right here. So we shared everything below that in this case. Similarly for X plus Y is less or equal to four is shown by the purple line, which will shade everything below again. So you want the area where both shaded regions overlap. We notice here, this is right around this area where both regions are overlapping. That's so we can find the graph that has this shaded area. See graphic A does not look like the same shaded area. Graff and graff. C. Does not, and d is no solution, which we can tell there is a solution. We can tell them that our solution is graph Beak. Okay, I hope to help you solve the problem. Thank you for watching. Goodbye

7. Systems of Equations & Matrices

Graphing Systems of Inequalities

College Algebra

7. Systems of Equations & Matrices

Graphing Systems of Inequalities

1:26 minutes

Problem 27

Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}3x + 6y \leq 6 \\2x + y \leq 8\end{cases}$

Verified step by step guidance

Step 1: Identify the system of inequalities to graph: \[\left\{ \begin{array}{l} 2x + 4y \leq 8 \\ 4x + y \leq 4 \end{array} \right.\]

Step 2: Convert each inequality into an equation to find the boundary lines: For the first inequality: \[2x + 4y = 8\] For the second inequality: \[4x + y = 4\]

Step 3: Find the intercepts for each boundary line to help graph them: - For \[2x + 4y = 8\]: - When \[x=0\], solve for \[y\]: \[4y = 8 \Rightarrow y = 2\] - When \[y=0\], solve for \[x\]: \[2x = 8 \Rightarrow x = 4\] - For \[4x + y = 4\]: - When \[x=0\], solve for \[y\]: \[y = 4\] - When \[y=0\], solve for \[x\]: \[4x = 4 \Rightarrow x = 1\]

Step 4: Graph the boundary lines using the intercepts found. Since the inequalities are $ \leq $, shade the region below or on each line. To determine which side to shade, pick a test point (usually the origin $0,0$) and check if it satisfies the inequality.

Step 5: The solution set to the system is the intersection of the shaded regions from both inequalities. This overlapping region represents all points $(x,y)$ that satisfy both inequalities simultaneously.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 possible solutions to 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 both inequalities on the same coordinate plane reveals the common shaded area representing all solutions.

Finding Boundary Lines and Test Points

To graph inequalities, first convert each inequality to an equation to find the boundary line. Then, select a test point not on the line (commonly the origin) to determine which side of the line satisfies the inequality. This method ensures accurate shading of the solution region.

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