Questions: Solve each system of inequ x-y ≤ 2 x+2y ≥ 1

Solution Steps

To solve the system of inequalities, we need to find the region in the coordinate plane that satisfies both inequalities simultaneously. We can do this by plotting the boundary lines of each inequality and then determining the region that satisfies both conditions.

We start with the system of inequalities: \[ \begin{array}{l} x - y \leq 2 \\ x + 2y \geq 1 \end{array} \]

Convert the inequalities to equations to find the boundary lines: \[ \begin{array}{l} x - y = 2 \\ x + 2y = 1 \end{array} \]

Solve each equation for \( y \): \[ \begin{array}{l} y = x - 2 \\ y = \frac{1 - x}{2} \end{array} \]

Plot the lines \( y = x - 2 \) and \( y = \frac{1 - x}{2} \) to visualize the boundaries.

Identify the region that satisfies both inequalities:

• For \( x - y \leq 2 \), the region is below the line \( y = x - 2 \).
• For \( x + 2y \geq 1 \), the region is above the line \( y = \frac{1 - x}{2} \).

The feasible region is the intersection of these two regions.

Final Answer