Question

Graph the system of linear inequalities. \(x+1>y\) \(y \geq 0\)

Short answer

The resulting graph will show the overlapping area which is the required solution for the system of given inequalities.

Step-by-step solution

Graph the first inequality

Plot the line for the inequality \(x+1=y\) on the graph. Since there is a 'greater than' sign, use a dashed line. As per the inequality, points above this line will satisfy the inequality, so shade everything above the line. This gives the solution set for the first inequality.

Graph the second inequality

Now, plot the line for the inequality \(y \geq 0\). Since there is a 'greater than or equal to' sign, use a solid line. According to the inequality, points above this line will satisfy it so, shade everything above the line. This gives the solution set for the second inequality.

Find the common solution area

The combined graph for both the inequalities will show an overlap area. This overlapping area is the solution to the system, as it satisfies both the inequalities simultaneously.

Key concepts

System of Linear Inequalities

Understanding a system of linear inequalities is crucial for solving many algebraic problems. Such a system consists of two or more linear inequalities that are considered together. The goal is to find the solution set, which represents all the points that satisfy all inequalities in the system simultaneously. For example, consider the inequalities from our exercise, \(x+1>y\)
and
\(y \geq 0\).
Individually, these inequalities describe regions on a graph. When combined, they form a system, and the solution for this system is the area where the individual solutions overlap. This intersection represents the set of all (x, y) coordinate pairs that make both inequalities true simultaneously. It's akin to finding common ground between two parties, where each party's requirements are represented by one inequality in the system.

Inequality Graphing

The process of inequality graphing involves visually representing the solution set of inequalities on a coordinate plane. Here's how you can effectively graph inequalities step by step:

• Begin by transforming the inequality into an equality and graphing the corresponding line.
• If the inequality is strict (using 'greater than' or 'less than'), use a dashed line to indicate that points on the line are not included in the solution set.
• If the inequality is not strict (using 'greater than or equal to' or 'less than or equal to'), draw a solid line, including the points on the line as part of the solution set.
• Use a shade or color to represent the area of the graph that satisfies the inequality—generally above the line for 'y greater than' inequalities and below for 'y less than'.

This graphical method highlights the regions that meet the criteria set by each inequality. By following these steps, students can accurately determine the solution set for each inequality before seeking the common solution area.

Solution Set of Inequalities

The solution set of inequalities is a collection of all the ordered pairs that satisfy all the inequalities in a given system. When it comes to visualizing the solution set on a graph, the intersection of all the shaded areas indicates where the solutions lie. In our exercise, the region that satisfies both \(x+1>y\)
and
\(y \geq 0\)
is the solution set. To identify this region, one must look for where the shaded areas corresponding to each inequality overlap. Once the overlap is determined, any point within this common region is a part of the solution set. It's essential to note that in a system with more than one inequality, points that satisfy only one of the inequalities are not part of the final solution set. In practice, graphing the solution set provides a clear visual and helps to grasp the concept of simultaneous solutions – where multiple conditions need to be met at once.