Questions: Graph the solution set. If there is no solution, indicate that the solution set is ∅. 3x + y ≤ 2 -5x + 2y ≥ 4

Solution Steps

The given inequalities are:

\(3x + y \le 2\) \(-5x + 2y \ge 4\)

Rewrite the first inequality in slope-intercept form: \(y \le -3x + 2\)

Rewrite the second inequality in slope-intercept form: \(2y \ge 5x + 4\) \(y \ge \frac{5}{2}x + 2\)

First Inequality: \(y \le -3x + 2\)

• Plot the y-intercept, which is 2.
• Use the slope, -3, to find another point on the line (e.g., move down 3 units and right 1 unit).
• Draw a solid line since the inequality includes the equal sign.
• Shade the region below the line since it is a "less than or equal to" inequality.

Second Inequality: \(y \ge \frac{5}{2}x + 2\)

• Plot the y-intercept, which is 2.
• Use the slope, 5/2, to find another point on the line (e.g., move up 5 units and right 2 units).
• Draw a solid line since the inequality includes the equal sign.
• Shade the region above the line since it is a "greater than or equal to" inequality.

The solution set is the region where the shaded areas of both inequalities overlap. In this case, the lines intersect at the point (0, 2). The solution set is the region above the line \(y = \frac{5}{2}x + 2\) and below the line \(y = -3x + 2\).

Final Answer