Graph the solution set of each system of inequalities or indicate that the system has no solution. { 2 x − 5 y ≤ 10 3 x − 2 y > 6 \begin{cases}2x - 5y \leq 10 \\3x - 2y > 6\end{cases} { 2 x − 5 y ≤ 10 3 x − 2 y > 6

Consider the following system of inequalities. X -4. Y is less than equal to eight and 4 X -2. Y is greater than 12. So you want to find the solution set the solutions that in the graph is given by the shaded region where both shaded areas overlap. Let's take a look at the graphic. Our first line is X -4, Y is less than equal to eight. This is given by our green line here. So everything above this green line Is going to be our solution set for X -4 Y is less than equal to eight. Similarly for the purple line, it will be our other equation for X -2, Y is greater than 12. Our solution set will be the area where both of these shaded regions overlap. We notice here that this happens right here in this shape of kind of a triangle. Let's see if you can find an answer that has this shaded area graph A does not have that shaded area graph. It also doesn't. But if you notice here, graph C looks like it has the shaded area, which is the answer to our problem is graph See Okay, hope to help you solve the problem. Thanks for watching. Goodbye

7. Systems of Equations & Matrices

Graphing Systems of Inequalities

College Algebra

7. Systems of Equations & Matrices

Graphing Systems of Inequalities

1:25 minutes

Problem 29

Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}2x - 5y \leq 10 \\3x - 2y > 6\end{cases}$

Verified step by step guidance

Step 1: Identify each inequality and rewrite them in slope-intercept form (y = mx + b) to make graphing easier. For the first inequality, start with $x - 4y \leq 8$. Solve for $y$ by isolating it on one side.

Step 2: For $x - 4y \leq 8$, subtract $x$ from both sides to get $-4y \leq -x + 8$. Then divide every term by $-4$. Remember to reverse the inequality sign when dividing by a negative number, resulting in $y \geq \frac{1}{4}x - 2$.

Step 3: For the second inequality, \$4x - 2y > 12$, isolate $y$ by subtracting $4x$ from both sides: $-2y > -4x + 12$. Then divide every term by $-2$, reversing the inequality sign to get $y < 2x - 6$.

Step 4: Graph the boundary lines $y = \frac{1}{4}x - 2$ (solid line because of \leq) and $y = 2x - 6$ (dashed line because of >). Then shade the region above the first line (since $y \geq$) and below the second line (since $y <$).

Step 5: The solution set is the overlapping shaded region that satisfies both inequalities. If there is no overlap, then the system has no solution.

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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 represented by the corresponding linear equation and then shading the region that satisfies the inequality. For '≤' or '≥', the boundary line is solid, indicating points on the line are included. For '<' or '>', the line is dashed, showing points on the line are excluded.

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. If no common region exists, the system has no solution.

Slope-Intercept Form and Boundary Lines

Converting inequalities to slope-intercept form (y = mx + b) helps in graphing by identifying the slope and y-intercept of the boundary line. This form makes it easier to draw the line accurately and determine which side to shade based on the inequality sign.

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