Answer:
see attached
Explanation:
The "slope-intercept" form of an equation for a line is ...
y = mx + b . . . . . . slope m, y-intercept b; m = rise/run
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boundary lines
There are several ways to graph equations for a line. One is to make use of the intercepts. That is easiest when their coordinates are integer values. The second inequality is in slope-intercept form, so it is easy to see that its y-intercept is -2.
The first inequality can be put into slope-intercept form by solving for y. That is easily done here by subtracting 6x:
y ≤ -6x +10
This shows you the boundary line has a slope of -6 and a y-intercept of 10. To graph it, you can plot the points (0, 10) and (1, 4) and draw a line through them. The point (1, 4) is found by using a "rise" of -6 and a "run" of 1 from the y-intercept:
m = -6 = rise/run = -6/1
(y-intercept) + (run, rise) = (0, 10) +(1, -6) = (1, 4) . . . . another point on the line
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Similarly, the boundary line of the second inequality goes through its y-intercept (0, -2) and the point (y-intercept) +(1, 5) = (1, 3). Points (0,-2) and (1, 3) are two points on that boundary line.
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In both cases, the inequality includes the "or equal to ..." case, so the boundary line is part of the solution set. Accordingly, each is drawn as a solid line (not dashed).
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shading
To determine where the shading is, you can choose any term with a positive coefficient and look at where it is in relation to the inequality symbol. For your two inequalities, we can look at the left side of the inequality.
first: 6x ≤
second: y ≥
These tell you the first solution set is shaded to the left of the boundary line (where x-values are less than those on the line). The second solution set is shaded above the boundary line (where y-values are greater than those on the line). The solution set for the system of inequalities is the doubly-shaded area to the left of both boundary lines.
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Additional comment
The x-term in the second inequality has a positive coefficient also. If we were to look at that one, we would see ≥ 5x. That is, the boundary line has greater x-values than those in the solution set, so it, too, will be shaded to the left. (Left and above are the same half-plane for a line with positive slope.)
If both coefficients are negative, you can multiply by -1, which will also reverse the inequality symbol.
Example: -x-y>6 ⇒ x+y<-6; shaded left or below dashed line.