Answer:
see the attached graph
Explanation:
You want to graph these inequalities and identify their solution space.
Boundary lines
The boundary line associated with the solution of an inequality is found by replacing the inequality symbol with an equal sign. Here, that means the boundary lines are given by the equations ...
These lines can be plotted by finding their x- and y-intercepts, then drawing the line through those points. In each case, the intercept is found by setting the other variable to zero and solving the resulting equation.
x + y ≤ 8
x-intercept of x+y=8: x = 8, or point (8, 0)
y-intercept of x+y=8: y = 8, or point (0, 8)
The inequality symbol for this inequality is "less than or equal to", so the boundary line is included in the solution set. That means the line is drawn as a solid (not dashed) line.
When we look at one of the variables with a positive coefficient, we see ...
x ≤ ... — shading is to the left of the boundary line
or
y ≤ ... — shading is below the boundary line
The solution space for this inequality is shown in blue in the attached graph.
x - y ≤ 2
The x- and y-intercepts are found the same way as above. They are ...
x-intercept: x = 2, or point (2, 0)
y-intercept: y = -2, or point (0, -2)
The boundary line is solid, and shading is to its left:
x ≤ ...
The solution space for this inequality is shown in red in the attached graph.
Solution space
The solutions of the set of inequalities are all the points on the graph where the shaded areas overlap. This is the left quadrant defined by the X where the lines cross.
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Additional comment
To summarize the "step-by-step", you want to ...
- determine the type of boundary line (dashed [<>], solid [≤≥])
- graph the boundary line using any convenient method
- determine the direction of shading, and shade the solution space
In this process, you make use of your knowledge of plotting points and lines. You also make use of your understanding of "greater than" or "less than" relationships in the x- and y-directions on a graph.