Answered

Get the answers you need at Westonci.ca, where our expert community is dedicated to providing you with accurate information. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

Graph the solution to the system of inequalities on the coordinate plane:

1. [tex]\( y \ \textgreater \ 2x + 4 \)[/tex]
2. [tex]\( x + y \leq 6 \)[/tex]

Sagot :

GinnyAnswer
Let's solve this system of inequalities step by step and graph the solution on the coordinate plane.

### Step 1: Graph the Boundary Lines

First, we'll graph the boundary lines for each inequality.

#### 1. Inequality [tex]\( y > 2x + 4 \)[/tex]

The boundary line for this inequality is:
[tex]\[ y = 2x + 4 \][/tex]

To graph this line, we need two points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) + 4 = 4 \][/tex]
So, the point [tex]\((0, 4)\)[/tex].

- When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 2(1) + 4 = 6 \][/tex]
So, the point [tex]\((1, 6)\)[/tex].

Plot the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 6)\)[/tex], and draw a dashed line through them (since the inequality is [tex]\( > \)[/tex], not [tex]\( \geq \)[/tex]).

#### 2. Inequality [tex]\( x + y \leq 6 \)[/tex]

The boundary line for this inequality is:
[tex]\[ x + y = 6 \][/tex]

To graph this line, we need two points:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 6 \][/tex]
So, the point [tex]\((0, 6)\)[/tex].

- When [tex]\( y = 0 \)[/tex]:
[tex]\[ x = 6 \][/tex]
So, the point [tex]\((6, 0)\)[/tex].

Plot the points [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex], and draw a solid line through them (since the inequality is [tex]\( \leq \)[/tex]).

### Step 2: Shade the Solution Areas

Now, we need to determine which regions to shade for each inequality.

#### 1. Inequality [tex]\( y > 2x + 4 \)[/tex]

Since it’s [tex]\( y > 2x + 4 \)[/tex], we shade the region above the line [tex]\( y = 2x + 4 \)[/tex].

#### 2. Inequality [tex]\( x + y \leq 6 \)[/tex]

Since it’s [tex]\( x + y \leq 6 \)[/tex], we shade the region below or on the line [tex]\( x + y = 6 \)[/tex].

### Step 3: Find the Intersection of the Shaded Areas

The solution to the system of inequalities will be the region where the shaded areas overlap. Here's what you should see on the graph:

1. The line [tex]\( y = 2x + 4 \)[/tex] (dashed) going through the points [tex]\((0, 4)\)[/tex] and [tex]\((1, 6)\)[/tex] with shading above this line.
2. The line [tex]\( x + y = 6 \)[/tex] (solid) going through the points [tex]\((0, 6)\)[/tex] and [tex]\((6, 0)\)[/tex] with shading below or on this line.

The overlapping shaded region represents the solution set to the system of inequalities.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.