In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x−y≤1, x≥2

Verified step by step guidance

1. Graph the line 2x - 3y = 6 by finding the x- and y-intercepts.
2. For the x-intercept, set y = 0 and solve for x: 2x = 6.
3. For the y-intercept, set x = 0 and solve for y: -3y = 6.
4. Shade the region below the line 2x - 3y = 6 to represent 2x - 3y ≤ 6.
5. Draw a vertical line at x = 5 and shade the region to the right to represent x ≥ 5.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They can be represented using symbols such as ≤ (less than or equal to) and ≥ (greater than or equal to). Understanding how to manipulate and graph inequalities is crucial for solving systems of inequalities, as it allows us to visualize the solution sets on a coordinate plane.

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line of the inequality and determining which side of the line represents the solution set. For a linear inequality in two variables, the line is dashed if the inequality is strict (e.g., < or >) and solid if it includes equality (e.g., ≤ or ≥). The area that satisfies the inequality is shaded, indicating all possible solutions.

Systems of Inequalities

A system of inequalities consists of two or more inequalities that share the same variables. The solution to the system is the region where the shaded areas of all inequalities overlap on the graph. Analyzing systems of inequalities is essential for determining feasible solutions in various applications, such as optimization problems in economics and engineering.