Graph the feasible region for the following system of inequalities. \( \begin{array}{l}x+y \leq 5 \\ x-y \geq 6\end{array} \)

We want all points (x, y) that satisfy both   (1) x + y ≤ 5  and  (2) x − y ≥ 6. Step 1. Rewrite the Inequalities in Slope‐Intercept Form • For (1), x + y ≤ 5 can be rewritten as y ≤ 5 − x.   This inequality represents all the points on or below the line     x + y = 5  (which has intercepts (0, 5) and (5, 0)). • For (2), x − y ≥ 6 can be rewritten as follows:   x − y ≥ 6  ⟹  –y ≥ 6 − x  ⟹ (multiplying each side by –1, which reverses the inequality)     y ≤ x − 6.   This inequality represents all the points on or below the line     y = x − 6  (which has intercepts (0, –6) and (6, 0)). Step 2. Find the Intersection of the Boundary Lines The two lines are:   Line A: x + y = 5   Line B: y = x − 6 Find their intersection by substituting the expression for y from Line B into Line A:   x + (x − 6) = 5  ⟹  2x − 6 = 5  ⟹  2x = 11  ⟹  x = 11/2. Then,   y = (11/2) − 6 = (11 − 12)/2 = −1/2. Thus, the lines intersect at (11/2, –1/2). Step 3. Graphing the Boundaries and Shading the Feasible Region 1. Draw the line x + y = 5.   • Plot the intercepts (0, 5) and (5, 0).   • Since the inequality is y ≤ 5 − x, shade the region below this line. 2. Draw the line y = x − 6.   • Plot the intercepts (0, –6) and (6, 0).   • Since the inequality is y ≤ x − 6, shade the region below this line. 3. The feasible region is the intersection of these two shaded areas. That is, it consists of all points that lie on (or below) both lines. Step 4. Describe the Feasible Region The feasible region is an unbounded area in the plane. Its boundary consists of:  • The line x + y = 5 (solid line because the inequality is “≤”)  • The line y = x − 6 (solid line because the inequality is “≤”) with the two regions below these lines intersecting. The two lines cross at (11/2, –1/2), which is the “corner point” of the feasible region. Step 5. Sketch (A Description for Your Graph) 1. Draw a set of axes. 2. Plot the point (0, 5) and (5, 0) to draw the line x + y = 5. 3. Plot the points (0, –6) and (6, 0) to draw the line y = x − 6. 4. Mark the intersection point (11/2, –1/2) ≈ (5.5, –0.5). 5. Shade the region that lies below the line x + y = 5. 6. Also shade the region that lies below the line y = x − 6. 7. The feasible region is the overlapping area from these two shadings. That is how you graph the feasible region for the given system of inequalities.