Problem 95 - Algebra and Trigonometry

Question

Explain how to graph the solution set of a system of inequalities.

Step-by-Step Solution

Answer

Graphing the solution set of a system of inequalities involves expressing the inequalities in slope-intercept form, graphing these lines using the slope and y-intercept, shading the appropriate side of each line according to the inequality sign, and identifying the area of overlap as the solution set.

Step 1: Express Inequalities in Slope-Intercept Form

Write each inequality in the system in slope-intercept form, y = mx + b, where m is the slope, and b is the y-intercept. This form will make the inequalities simple to graph as lines on a coordinate plane.

Step 2: Graph the Lines

Using the slope and y-intercept from Step 1, graph each line on the same coordinate plane. If the inequality is strict (meaning less than '<' or greater than '>'), draw a dashed line, which indicates that the points on the line are not included in the solution. If the inequality is not strict (meaning less than or equal to '≤' or greater than or equal to '≥'), draw a solid line, which indicates that the points on the line are included in the solution.

Step 3: Shade the Appropriate Areas

Then, determine which side of each line to shade. For a 'greater than' inequality, shade above the line, and for a 'less than' inequality, shade below the line. Use a different texture or color for each line.

Step 4: Identify Solution Set

The solution to the system of inequalities will be the area where the shadings for all the inequalities overlap. This area represents all possible solutions to the system of inequalities.