Westonci.ca connects you with experts who provide insightful answers to your questions. Join us today and start learning! Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
Sure, let's graph the inequality [tex]\( x + y \leq 1 \)[/tex] step-by-step.
### Step 1: Understand the Inequality
The inequality [tex]\( x + y \leq 1 \)[/tex] implies that we are looking for all pairs of [tex]\( (x, y) \)[/tex] that satisfy this condition. To graph this on the coordinate plane, we need to consider the equation [tex]\( x + y = 1 \)[/tex] as well.
### Step 2: Draw the Boundary Line
First, we draw the boundary line [tex]\( x + y = 1 \)[/tex]:
1. Find the intercepts.
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex]. This gives the point [tex]\( (0, 1) \)[/tex].
- When [tex]\( y = 0 \)[/tex], [tex]\( x = 1 \)[/tex]. This gives the point [tex]\( (1, 0) \)[/tex].
2. Plot these points on the coordinate plane:
- Point [tex]\( (0, 1) \)[/tex]
- Point [tex]\( (1, 0) \)[/tex]
3. Draw a straight line passing through these points. This line represents [tex]\( x + y = 1 \)[/tex].
### Step 3: Determine the Shading Region
Next, we need to determine which side of the line to shade for the inequality [tex]\( x + y \leq 1 \)[/tex]:
1. Choose a test point that is not on the line to determine where to shade. A good test point is [tex]\( (0, 0) \)[/tex].
- Substitute [tex]\( (0, 0) \)[/tex] into the inequality: [tex]\( 0 + 0 \leq 1 \)[/tex].
- Simplifies to [tex]\( 0 \leq 1 \)[/tex], which is true.
2. Since the test point [tex]\( (0, 0) \)[/tex] satisfies the inequality, we shade the region that includes [tex]\( (0, 0) \)[/tex]. This means we shade the region below and to the left of the line [tex]\( x + y = 1 \)[/tex].
### Step 4: Considering the Inequality
Since the inequality is [tex]\( \leq \)[/tex], the line [tex]\( x + y = 1 \)[/tex] is also part of the solution. Therefore, we keep the line solid (instead of dashed) to include points on the line itself.
### Step 5: Final Graph
Putting it all together:
- Draw the line [tex]\( x + y = 1 \)[/tex] passing through points [tex]\( (0, 1) \)[/tex] and [tex]\( (1, 0) \)[/tex].
- Shade the region below (and including) this line.
### Visual Representation
Here’s a sketch of the graph:
```
y
|
| /
| /
| /
|________/_______ x
(-10,11) *(0,1)
```
The shaded region includes the line [tex]\( x + y = 1 \)[/tex] and all points below it. The points might not be clear without a proper graphing tool, but you can visualize a straight line passing through [tex]\( (0, 1) \)[/tex] and [tex]\( (1, 0) \)[/tex], and shade everything below and on this line.
### Step 1: Understand the Inequality
The inequality [tex]\( x + y \leq 1 \)[/tex] implies that we are looking for all pairs of [tex]\( (x, y) \)[/tex] that satisfy this condition. To graph this on the coordinate plane, we need to consider the equation [tex]\( x + y = 1 \)[/tex] as well.
### Step 2: Draw the Boundary Line
First, we draw the boundary line [tex]\( x + y = 1 \)[/tex]:
1. Find the intercepts.
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex]. This gives the point [tex]\( (0, 1) \)[/tex].
- When [tex]\( y = 0 \)[/tex], [tex]\( x = 1 \)[/tex]. This gives the point [tex]\( (1, 0) \)[/tex].
2. Plot these points on the coordinate plane:
- Point [tex]\( (0, 1) \)[/tex]
- Point [tex]\( (1, 0) \)[/tex]
3. Draw a straight line passing through these points. This line represents [tex]\( x + y = 1 \)[/tex].
### Step 3: Determine the Shading Region
Next, we need to determine which side of the line to shade for the inequality [tex]\( x + y \leq 1 \)[/tex]:
1. Choose a test point that is not on the line to determine where to shade. A good test point is [tex]\( (0, 0) \)[/tex].
- Substitute [tex]\( (0, 0) \)[/tex] into the inequality: [tex]\( 0 + 0 \leq 1 \)[/tex].
- Simplifies to [tex]\( 0 \leq 1 \)[/tex], which is true.
2. Since the test point [tex]\( (0, 0) \)[/tex] satisfies the inequality, we shade the region that includes [tex]\( (0, 0) \)[/tex]. This means we shade the region below and to the left of the line [tex]\( x + y = 1 \)[/tex].
### Step 4: Considering the Inequality
Since the inequality is [tex]\( \leq \)[/tex], the line [tex]\( x + y = 1 \)[/tex] is also part of the solution. Therefore, we keep the line solid (instead of dashed) to include points on the line itself.
### Step 5: Final Graph
Putting it all together:
- Draw the line [tex]\( x + y = 1 \)[/tex] passing through points [tex]\( (0, 1) \)[/tex] and [tex]\( (1, 0) \)[/tex].
- Shade the region below (and including) this line.
### Visual Representation
Here’s a sketch of the graph:
```
y
|
| /
| /
| /
|________/_______ x
(-10,11) *(0,1)
```
The shaded region includes the line [tex]\( x + y = 1 \)[/tex] and all points below it. The points might not be clear without a proper graphing tool, but you can visualize a straight line passing through [tex]\( (0, 1) \)[/tex] and [tex]\( (1, 0) \)[/tex], and shade everything below and on this line.
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
