Question

Graph the system of linear inequalities. \(x \geq 2\) \(x-2 y \geq 3\) \(3 x+y \geq 9\) \(x+y \leq 7\)

Short answer

The solution to the system of inequalities is the overlapped region in the graph where all shaded regions of the four inequalities converge.

Step-by-step solution

Graph the first inequality

The first inequality is \(x \geq 2\). This is a vertical line crossing the x-axis at 2. Because of the 'greater than or equal to' symbol, the line will be solid, and the region of the graph to the right of the line will be shaded.

Graph the second inequality

The second inequality is \(x-2y \geq 3\). It's standard form is \(y \leq (1/2)x -3/2\). This will be a line that is increasing with positive slope \(1/2\) and cuts y-axis at -3/2. The line will be solid, and the area under the line will be shaded. Overlay this on the previous graph.

Graph the third inequality

The third inequality is \(3x+y \geq 9\). Transform this into slope-intercept form: \(y \leq -3x +9\). This line has a negative slope and cuts y-axis at 9. For this inequality, the line is solid and the area under the line is shaded. Overlay this on the previous graph.

Graph the fourth inequality

The fourth and final inequality is \(x+y \leq 7\). Transform to slope-intercept form: \(y \leq -x +7\). This line falls downward and cuts y-axis at 7. The line itself is solid and the area below the line is to be shaded. Overlay this line on the graph with the previous lines.

Find the overlapped region

The overlapped region of the graph, where all shaded regions meet, is the solution to the system of equations. This area represents all possible combinations of x and y which satisfy all the given inequalities simultaneously.