In a system of inequalities, the choice of solid or dotted boundary lines depends on whether the boundary itself is included in the solution set. The symbol used in the inequality determines this:
Solid Line (≤ or ≥): A solid line is used when the boundary is included in the solution set. This means that points lying on the line satisfy the inequality.
Dotted Line (< or >): A dotted line is used when the boundary is not included in the solution set. In this case, points on the line do not satisfy the inequality.
Now, let's look at the inequalities provided:
Inequality #1: y≤x+4
Since it includes the "or equal to" part (≤), a solid line is appropriate.
Inequality #2: y≤x+1
Similarly, it includes the "or equal to" part, so a solid line is used.
For the graphing part:
Draw a solid line for Inequality #1 (y=x+4).
Draw another solid line for Inequality #2 (y=x+1).
These lines represent the boundaries of the solution set. Now, shading is done based on the direction of the inequalities. Since both inequalities have y on the left side, the shaded region is determined by where y is less than or equal to the expressions on the right side.
Shade the region below both lines because it represents the points where y is less than or equal to both x+4 and x+1. This shaded region is the solution set for the given system of inequalities.
Ensure that the shading is consistent, and the final graph accurately represents the solution set.