Question

Graph the system of linear inequalities. \(2 x+y \geq 2\) \(x \leq 3\) \(y \leq \frac{1}{2}\)

Short answer

The solution to the system of linear inequalities is the intersecting region that satisfies all three inequalities: \(2x + y \geq 2\), \(x \leq 3\), and \(y \leq 0.5\).

Step-by-step solution

Graph each inequality individually

We start by plotting each inequality individually. The inequality \(2x + y \geq 2\) can be rewritten in slope-intercept form \(y = -2x + 2\). When graphed, this will be a line that intersects the y-axis at \(y = 2\) and has a downward slope of -2. As the inequality is 'greater than or equal to', the region above the line will be shaded. The inequality \(x \leq 3\) is a vertical line crossing the x-axis at \(x = 3\). As this is 'less than or equal to', the area to the left of the line will be shaded. Similarly, the inequality \(y \leq 0.5\) is a horizontal line crossing the y-axis at \(y = 0.5\). Here, the area below the line will be shaded because the inequality is 'less than or equal to'.

Identify the intersection

To identify the solution to the system of inequalities, locate the region where all shaded areas intersect. This region is the one that fulfills all three inequalities simultaneously.

Draw the final graph

Finally, draw the final graph highlighting the intersecting region. This is the solution to the problem.