Answer: To solve this problem, we need to graph the three inequalities and find the overlapping region.
First, let's graph the inequality y ≥ x - 4. We can start by graphing the line y = x - 4, which has a y-intercept of -4 and a slope of 1.
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Since we want the region where y is greater than or equal to x - 4, we shade the area above the line.
Next, let's graph the inequality y ≤ 10. This is a horizontal line passing through y = 10.
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Since we want the region where y is less than or equal to 10, we shade the area below the line.
Finally, let's graph the inequality 5 ≤ x + y. This is a line with a y-intercept of 5 and a slope of -1.
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10| +----+
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0 1 2 3 4 5
Since we want the region where x + y is greater than or equal to 5, we shade the area above the line.
Now we can find the overlapping region of the three shaded areas:
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0 1 2 3 4 5
The region is a triangle with vertices at (0, 4), (1, 5), and (5, 0).
To find the area of the triangle, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
The base of the triangle is the distance between the points (0, 4) and (5, 0), which is 5.
The height of the triangle is the distance between the point (1, 5) and the line 5 = x + y. We can find the equation of the line perpendicular to 5 = x + y and passing through (1, 5). This line has a slope of 1 and passes through (1, 5), so its equation is y = x + 4. We can find the intersection of this line and the line 5 = x + y by solving the system of equations:
y = x + 4
y = 5 - x
Substituting y = x + 4 into the second equation, we get:
x + 4 = 5 - x
Solving for x, we get:
x = 1
Explanation: