Get the answers you need from a community of experts on IDNLearn.com. Join our community to receive prompt, thorough responses from knowledgeable experts.
Sagot :
Sure, let's solve the given system of inequalities step-by-step:
Given:
1. [tex]\( x - y > -13 \)[/tex]
2. [tex]\( 2x + y > 0 \)[/tex]
Our objective is to find the region on the coordinate plane that satisfies both inequalities. Here's how we approach solving them:
### Step 1: Convert inequalities to equalities to find boundary lines
1. [tex]\( x - y = -13 \)[/tex]
2. [tex]\( 2x + y = 0 \)[/tex]
### Step 2: Plot the boundary lines
#### Line 1: [tex]\( x - y = -13 \)[/tex]
To graph this line, we find the intercepts:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ 0 - y = -13 \implies y = 13 \][/tex]
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ x - 0 = -13 \implies x = -13 \][/tex]
So, the line passes through points [tex]\((0, 13)\)[/tex] and [tex]\((-13, 0)\)[/tex].
#### Line 2: [tex]\( 2x + y = 0 \)[/tex]
To graph this line, we also find the intercepts:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ 2(0) + y = 0 \implies y = 0 \][/tex]
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ 2x + 0 = 0 \implies x = 0 \][/tex]
In this case, this reduces to [tex]\( y = -2x \)[/tex].
### Step 3: Determine the regions for the inequalities
#### For [tex]\( x - y > -13 \)[/tex]:
Choose a test point not on the line, say [tex]\( (0,0) \)[/tex]:
[tex]\[ 0 - 0 > -13 \implies 0 > -13 \][/tex]
This is true, so the region above the line [tex]\( x - y = -13 \)[/tex] (including the line extending to the right beyond [tex]\(0, 13\)[/tex] and below to the left of [tex]\((-13,0)\)[/tex]) satisfies the inequality.
#### For [tex]\( 2x + y > 0 \)[/tex]:
Again, use the test point [tex]\( (0,0) \)[/tex]:
[tex]\[ 2(0) + 0 > 0 \implies 0 > 0 \][/tex]
This is false; thus, the region above the line [tex]\( y = -2x \)[/tex] or [tex]\(2x + y = 0\)[/tex] (excluding the axis's intersection points and all points below/below-the-line).
### Step 4: Find the intersection region
The solution to the system of inequalities is where both conditions are true. This will be the overlapping region:
1. The region above the line [tex]\( x - y = -13 \)[/tex]
2. The region above the line [tex]\( 2x + y = -0 \)[/tex]or[tex]\( y > -2x \)[/tex]
### Step 5: Graphical representation
Now, plot both lines and shade the regions:
- Line 1: Dashed because it's not included, graph starts at [tex]\( (0,13) \)[/tex] and runs through [tex]\((-13,0)\)[/tex]
- Line 2: Dashed as it's not included, downward-sloped from [tex]\( y = -2x \)[/tex].
Finally, the intersection/visible region where both inequalities overlap represent points verifying both inequalities.
### Conclusion
The required region on the coordinate plane represents points where both inequalities align, described mathematically as:
- [tex]\( x - y > -13 \)[/tex]
- [tex]\( 2x + y > 0 \)[/tex]
Thus, our solution is the set of points that simultaneously satisfy these conditions where both shaded and dashed areas intersect.
Given:
1. [tex]\( x - y > -13 \)[/tex]
2. [tex]\( 2x + y > 0 \)[/tex]
Our objective is to find the region on the coordinate plane that satisfies both inequalities. Here's how we approach solving them:
### Step 1: Convert inequalities to equalities to find boundary lines
1. [tex]\( x - y = -13 \)[/tex]
2. [tex]\( 2x + y = 0 \)[/tex]
### Step 2: Plot the boundary lines
#### Line 1: [tex]\( x - y = -13 \)[/tex]
To graph this line, we find the intercepts:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ 0 - y = -13 \implies y = 13 \][/tex]
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ x - 0 = -13 \implies x = -13 \][/tex]
So, the line passes through points [tex]\((0, 13)\)[/tex] and [tex]\((-13, 0)\)[/tex].
#### Line 2: [tex]\( 2x + y = 0 \)[/tex]
To graph this line, we also find the intercepts:
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ 2(0) + y = 0 \implies y = 0 \][/tex]
- When [tex]\( y = 0 \)[/tex]:
[tex]\[ 2x + 0 = 0 \implies x = 0 \][/tex]
In this case, this reduces to [tex]\( y = -2x \)[/tex].
### Step 3: Determine the regions for the inequalities
#### For [tex]\( x - y > -13 \)[/tex]:
Choose a test point not on the line, say [tex]\( (0,0) \)[/tex]:
[tex]\[ 0 - 0 > -13 \implies 0 > -13 \][/tex]
This is true, so the region above the line [tex]\( x - y = -13 \)[/tex] (including the line extending to the right beyond [tex]\(0, 13\)[/tex] and below to the left of [tex]\((-13,0)\)[/tex]) satisfies the inequality.
#### For [tex]\( 2x + y > 0 \)[/tex]:
Again, use the test point [tex]\( (0,0) \)[/tex]:
[tex]\[ 2(0) + 0 > 0 \implies 0 > 0 \][/tex]
This is false; thus, the region above the line [tex]\( y = -2x \)[/tex] or [tex]\(2x + y = 0\)[/tex] (excluding the axis's intersection points and all points below/below-the-line).
### Step 4: Find the intersection region
The solution to the system of inequalities is where both conditions are true. This will be the overlapping region:
1. The region above the line [tex]\( x - y = -13 \)[/tex]
2. The region above the line [tex]\( 2x + y = -0 \)[/tex]or[tex]\( y > -2x \)[/tex]
### Step 5: Graphical representation
Now, plot both lines and shade the regions:
- Line 1: Dashed because it's not included, graph starts at [tex]\( (0,13) \)[/tex] and runs through [tex]\((-13,0)\)[/tex]
- Line 2: Dashed as it's not included, downward-sloped from [tex]\( y = -2x \)[/tex].
Finally, the intersection/visible region where both inequalities overlap represent points verifying both inequalities.
### Conclusion
The required region on the coordinate plane represents points where both inequalities align, described mathematically as:
- [tex]\( x - y > -13 \)[/tex]
- [tex]\( 2x + y > 0 \)[/tex]
Thus, our solution is the set of points that simultaneously satisfy these conditions where both shaded and dashed areas intersect.
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
