Question

Graph the system of linear inequalities. \(x \geq 1\) \(x-2 y \leq 3\) \(3 x+2 y \geq 9\) \(x+y \leq 6\)

Short answer

The solution to the system of inequalities is the region of overlap from the four individual shaded regions of each inequality on the graph.

Step-by-step solution

Graphing first inequality

The inequality \(x \geq 1\) will produce a vertical line on x=1. Shade the region to the right of the line because the region represents x greater than or equal to 1.

Graphing second inequality

To graph the inequality \(x-2 y \leq 3\), you will first write it in slope-intercept form: \(y \geq (1/2)x - 3/2\). The slope of the line is 1/2 and the y-intercept is -3/2. Graph the line, then shade the region below the line because the inequality symbol is less than or equal to.

Graphing third inequality

For the third inequality \(3 x+2 y \geq 9\), rewrite it in slope-intercept form like so: \(y \leq (-3/2)x + 9/2\). The slope of the line is -3/2 and the y-intercept is 9/2. Graph the line and shade the region below the line because the symbol in the inequality is a greater than or equal.

Graphing fourth inequality

The inequality \(x + y \leq 6\) needs to be written in slope intercept form, yielding \(y \leq -x + 6\). The slope of the line is -1 and y-intercept is 6. Graph this line and shade the region below the line because the inequality symbol is less than or equal to.

Find the overlap region

Now, the region that satisfies all inequalities is the one where all the shaded regions from previous steps overlap. This overlapping region is the solution to the system of inequalities.