Please simplify this and make it easier to understand.

Question

Day 1: Directions for Graphing Systems of Inequalities A system of inequalities consists of two inequalities to be solved together. You have learned that the solution of a linear inequality is represented graphically as a shaded region. The solution of a system of linear inequalities is the region where the shaded regions that represent the solutions of the inequalities overlap. The points in this region are the solutions that make the inequalities in the system true. When graphing systems of inequalities, follow the steps you've learned previously to graph each inequality. For example, to solve the system containing the inequalities and , graph each inequality, determining whether the boundary line should be solid or dashed, and shade the appropriate side of the boundary line. Remember that the boundary is a solid line if the inequality is or , and a dashed line if the inequality is or . When deciding which side to shade, remember to use a test point. If the resulting statement is true, shade the side of the line that includes the test point. If the resulting statement is false, shade the side that does not include the test point. In the graph below, the top shaded region is the solution of and the bottom shaded region is the solution of . The region where the shading overlaps (on the left) is the solution of the system of inequalities. Also included in this system’s solution are points on the solid boundary line. The x-axis and y-axis are numbered negative 5 to 5. A dotted line is plotted at points negative 2 comma 0 and 0 comma 1. The graph above the dotted line is shaded. A solid line is plotted at points 0 comma 1 and 5 comma negative 1. The graph below the solid line is shaded. The area of overlap is shaded darker. Identifying a System of Inequalities If you are given the graph of a system of inequalities, you can determine the inequalities in the system. First, y ou must determine the equation of the boundary line for each inequality. To do this, find the y-intercept and the slope and write the corresponding equation. Suppose you are given the following system: The x-axis and y-axis are numbered negative 5 to 5. A dotted line is plotted at points negative 1 comma 0 and 0 comma 1. The graph above the dotted line is shaded. A solid line is plotted at points 0 comma 1 and 1 comma 1. The graph below the solid line is shaded. The area of overlap is shaded darker. Both boundary lines have a y-intercept of 1. The line with a positive slope has a slope of 1 and the line with the negative slope has a slope of . So, the equations corresponding to these boundary lines are and . With the boundary line of , the top region is shaded, so this is either or . Since the line is dashed, it is . With the boundary line of , the bottom region is shaded, so this is either or . Since it is solid, it is . Using these steps, the system of inequalities is established. The final answer is and . A point can be substituted in from the shaded region to make sure that it creates a true statement in the inequalities. Take the point . (true) (true)

Answer 1

A system of inequalities means there are two inequalities that need to be solved together. When we graph these inequalities, the solution is the overlapping shaded region. We can determine which side of the boundary line to shade by using a test point. If the point makes the inequality true, we shade that side. If the point makes the inequality false, we shade the other side. In the given example, the system of inequalities is graphed with a dotted line and a solid line. The overlapping shaded region is the solution. To identify the inequalities in a system given a graph, we find the y-intercept and slope of each boundary line. We can then write the corresponding equation. By using these steps, we can establish the system of inequalities. The final answer is found by substituting a point from the shaded region into the inequalities to see if it creates a true statement.