Final answer:
To graph a system of linear inequalities, write each inequality in slope-intercept form, then plot the lines and shade the solution region. The slope gives the steepness, and the y-intercept is where the line crosses the y-axis. The solution region is where all shaded areas from each inequality overlap.
Step-by-step explanation:
Graphing Systems of Linear Inequalities
Graphing a system of linear inequalities involves plotting the corresponding lines on a graph and then shading the region that represents the solution set. To graph the inequalities, each inequality needs to be written in slope-intercept form, which is y = mx + b where m is the slope and b is the y-intercept. The slope is a measure of how steep the line is, and the y-intercept is the point where the line crosses the y-axis. For example, with a slope of 3, the line rises 3 units vertically for every 1 unit it moves horizontally. Once the inequalities are in slope-intercept form, plot each line on the graph, using dashed or solid lines based on whether the inequality is strict (< or >) or includes equality (≤ or ≥).
To identify the solution region, test a point not on the line (often the origin, if it's not on the line) to see which side of the line it lies on. The side where the test point satisfies the inequality is the side that you will shade. When graphing more than one inequality, the solution region is typically the overlapped shaded area that satisfies all inequalities.