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.
