To graph the solution set of the first linear inequality [tex]\( y \leq x + 2 \)[/tex], follow these steps:
### Step-by-Step Solution:
1. Identify the Boundary Line Equation:
The inequality [tex]\( y \leq x + 2 \)[/tex] represents a region in the coordinate plane. To find the boundary line, we consider the equation [tex]\( y = x + 2 \)[/tex].
2. Determine the Type of Boundary Line:
Since the inequality is [tex]\( y \leq x + 2 \)[/tex], the boundary line [tex]\( y = x + 2 \)[/tex] will be solid, indicating that points on the line are included in the solution set.
- Choose the type of boundary line: Solid ( - )
3. Find Two Points on the Boundary Line:
To plot the line, we need at least two points. Here, we will use:
- When [tex]\( x = -10 \)[/tex] then [tex]\( y = -10 + 2 = -8 \)[/tex] — giving us Point 1: [tex]\((-10, -8)\)[/tex].
- When [tex]\( x = 10 \)[/tex] then [tex]\( y = 10 + 2 = 12 \)[/tex] — giving us Point 2: [tex]\((10, 12)\)[/tex].
- Enter two points on the boundary line: [tex]\((-10, -8)\)[/tex] and [tex]\((10, 12)\)[/tex].
4. Draw the Boundary Line:
Plot the points [tex]\((-10, -8)\)[/tex] and [tex]\((10, 12)\)[/tex] on the coordinate plane. Connect these points with a solid line since the boundary is inclusive.
5. Shade the Region:
The inequality [tex]\( y \leq x + 2 \)[/tex] indicates the region where the [tex]\( y \)[/tex]-values are less than or equal to the [tex]\( y \)[/tex]-values on the line [tex]\( y = x + 2 \)[/tex]. Therefore, shade the region below the line (and including the line itself).
### Graph Visualization:
- The boundary line [tex]\( y = x + 2 \)[/tex] will pass through the points [tex]\((-10, -8)\)[/tex] and [tex]\((10, 12)\)[/tex] and extend infinitely in both directions, forming a straight solid line.
- The region below and including the line [tex]\( y = x + 2 \)[/tex] will be shaded to represent all points [tex]\((x, y)\)[/tex] that satisfy [tex]\( y \leq x + 2 \)[/tex].
