Problem 13 - Finite Mathematics and Applied Calculus

Question

Sketch the region that corresponds to the given inequalities, say whether the region is bounded or unbounded, and find the coordinates of all corner points (if any). $$ \begin{array}{r} 4 x-y \leq 8 \\ x+2 y \leq 2 \end{array} $$

Step-by-Step Solution

Answer

The region corresponding to the given inequalities is bounded, and it has one corner point at the coordinates \(\(\frac{8}{9}, \frac{16}{9}\)\).

Step 1: Rewrite inequalities as equations

Rewrite the inequalities as equations to find the boundary lines: $$ \begin{array}{r} 4x-y = 8\\ x+2y = 2 \end{array} $$

Step 2: Plot the boundary lines

Now, plot the boundary lines on a coordinate plane. For the first equation, if \(x=0\), then \(y=-8\). If \(y=0\), then \(x=2\). For the second equation, if \(x=0\), then \(y=1\). If \(y=0\), then \(x=2\). Connect the points to draw the lines.

Step 3: Shade the regions that satisfy the inequalities

To determine which regions to shade, choose a test point (such as \((0,0)\)) and check if it satisfies both inequalities: $$ 4(0)-0 \leq 8 \ (\text{True}) \\ 0 +2(0) \leq 2 \ (\text{True}) $$ Since \((0,0)\) satisfies both inequalities, we shade the region containing \((0,0)\).

Step 4: Determine if the region is bounded or unbounded

Since the shaded region is enclosed by the two boundary lines, the region is bounded.

Step 5: Find the coordinates of all corner points

To find the coordinates of all corner points, solve the system of equations formed by the boundary lines: $$ \begin{array}{r} 4x-y = 8\\ x+2y = 2 \end{array} $$ Multiply the second equation by 4 to eliminate \(x\): $$ \begin{array}{r} 4x-y = 8\\ 4(x+2y) = 4(2) \end{array} $$ This yields: $$ \begin{array}{r} 4x-y = 8\\ 4x+8y = 8 \end{array} $$ Now, subtract the first equation from the second: $$ (4x + 8y) - (4x - y) = 8 - (-8) $$ This simplifies to: $$ 9y = 16 $$ Divide by 9: $$ y = \frac{16}{9} $$ Now, plug this value for \(y\) into the first equation: $$ 4x - \frac{16}{9} = 8 $$ Add \(\frac{16}{9}\) to both sides: $$ 4x = 8 + \frac{16}{9} $$ Multiply 8 by \(\frac{9}{9}\) to combine the fractions: $$ 4x = \frac{64}{9} $$ Finally, divide by 4 to find \(x\): $$ x = \frac{64}{36} = \frac{8}{9} $$ The coordinates of the corner point are \(\(\frac{8}{9}, \frac{16}{9}\)\).