Problem 6 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the graphical solution of each inequality. $$y \geq-1$$

Step-by-Step Solution

Answer

The short answer: To find the graphical solution of the inequality \(y \geq -1\), draw a solid horizontal line at y = -1 on the Cartesian plane. Then, shade the region above this line, including the line itself. This shaded region represents all the points that satisfy the inequality.

Step 1: Plot the boundary line

In this case, the boundary line is the horizontal line y = -1, which separates the region where the inequality is true and where it is not true. Draw the horizontal line y = -1 on the Cartesian plane, and make it a solid line as the inequality is "greater or equal to."

Step 2: Shade the solution region

The inequality states that y should be "greater or equal to" -1. This means the solution region would be the area where y values are greater than or equal to -1. Considering a few points above the line y = -1, you will find that these points satisfy the inequality. So, shade the region above the line y = -1, as it represents the solution of the inequality.

Step 3: Interpret the Graphical Solution

The graphical solution of the inequality \(y \geq -1\) is the entire region above the horizontal line y = -1 on the Cartesian Plane, including the line itself. Any point in this region, including the points on the line y = -1, is a solution to the given inequality.

Key Concepts

Plotting Inequalities

Understanding how to plot inequalities on a graph is a fundamental skill in algebra. Begin with a Cartesian plane, which includes a horizontal x-axis and a vertical y-axis. Inequalities can be visualized as areas on this plane. For the inequality \(y \geq -1\), you'll plot the horizontal line where \(y = -1\). This line acts as a reference point. Points on the line \(y = -1\) are included in the solution set since the inequality is 'greater than or equal to'.
To ensure accuracy when plotting, you should pick at least two points where \(y\) equals -1 and draw a straight line through them. Common points to use are \((0, -1)\) and \((1, -1)\). Remember, this line will be solid, not dashed, which indicates that points on the line are part of the solution.

Shading the Solution Region

The next step is identifying which side of the line to shade. Shading is a visual representation of all feasible solutions to the inequality. Because we are dealing with \(y \geq -1\), we are interested in every point where the y-value is equal to -1 or higher. Imagine taking a point above the line and checking the inequality: if the inequality holds true, that's where you shade. For \(y \geq -1\), you'll shade the entire region above the boundary line. This shaded area represents all the y-values that are greater than or minus one, visually capturing the set of all possible solutions.

Boundary Line in Inequalities

The boundary line has a crucial role in graphing inequalities. It serves as the divider between the region that satisfies the inequality and the one that does not. The nature of the inequality (whether it's 'greater than or equal to' or 'strictly greater than') determines if the line is solid or dashed. A solid line, such as in our example \(y \geq -1\), means that the line itself is included in the solution set. All points lying on the line \(y = -1\) are valid solutions. Conversely, if the inequality were 'greater than' (non-inclusive), the boundary line would be dashed to signify that points on the line are not part of the solution. It's a subtle but important distinction that visually communicates the nature of the inequality on the graph.