Problem 39 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}x-y \leq 1 \\\x \geq 2\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region right of the line \(x=2\) and below the line \(x-y=1\).

Step 1: Graphing Individual Inequalities

First, graph both inequalities on a cartesian coordinate system separately. The inequality x-y≤1 represents a straight line passing through (1, 0) and (0, -1). As the inequality is less than or equal to, all the points on the line and below the line will be the solution set for this inequality. Draw this on the graph. The inequality x≥2 is a vertical line passing through (2, 0). As the inequality is greater than or equal to, all the points on the line and to the right of the line are the solution set for this inequality. Also draw this on the graph.

Step 2: Combining Inequalities

The solution to the system of inequalities is the region of overlap on the graph or where both inequalities are true. On the graph, you will find a region to the right of the line x=2 and below the line x-y=1. This region is the solution to the system of inequalities.

Step 3: Indicating No Solution

If there was no region of overlap, then the system of inequalities would have no solution. But for this case, as there is an overlapping region, a solution exists.

Key Concepts

Graphing Inequalities

Graphing inequalities involves plotting areas of the graph where the inequality expressions hold true. To visualize an inequality on a graph, it is helpful to start by viewing it as an equation to plot the boundary line. Once the line is drawn, shading is used to represent all solutions of the inequality. For example, when graphing the inequality \(x - y \leq 1\), draw the line \(x - y = 1\). Because the inequality sign \(\leq\) includes equality, the line itself is part of the solution set, and you should use a solid line. Shade below this line to show all the points (solutions) that satisfy the inequality. This shaded region represents all the possible solutions to the inequality.

• Draw the line based on the boundary equation (equality).
• Use a solid line if the inequality sign is \(\leq\) or \(\geq\), and a dashed line if it is \(<\) or \(>\).
• Shade the area that satisfies the inequality.

Cartesian Coordinate System

The Cartesian Coordinate System is a mathematical tool that allows us to graph equations and inequalities. This system is composed of two perpendicular axes that intersect at a point called the origin, labeled \((0, 0)\). The horizontal axis is known as the x-axis, and the vertical axis is called the y-axis. Each point in this plane is defined by a pair of numbers called coordinates, \((x, y)\). These coordinates indicate how far the point is from the origin along each axis.

• The x-coordinate indicates the point's location horizontally.
• The y-coordinate indicates the point's location vertically.

When plotting inequalities, the Cartesian coordinate system helps by displaying the solution set visually, allowing us to easily identify the region of solutions or overlapping areas for systems of inequalities.

Solution Set

In the context of inequalities, the solution set is the set of all points that satisfy the given inequality or system of inequalities. For a single inequality, the solution set is often a shaded region on the graph.
When dealing with a system of inequalities, the solution set is where the shaded regions from each inequality overlap. This region represents the set of points that satisfy all the inequalities at once. When graphing the system with \(x - y \leq 1\) and \(x \geq 2\), the solution set is the shared area to the right of the \(x = 2\) line and below the \(x - y = 1\) line.

• Identify each inequality's shaded region on the graph.
• The intersecting area of the shaded regions represents the solution set.
• If there's no intersection, the system has no solution.

Linear Inequality

A linear inequality is very similar to a linear equation but involves instances of \(<\), \(>\), \(\leq\), or \(\geq\) instead of an equal sign. In a graph, a linear inequality defines a region of the coordinate plane as opposed to just a line.
The boundary line, determined by the corresponding linear equation, divides the plane into two halves: one that represents points that satisfy the inequality and one that does not. For example, \(x - y \leq 1\) translates to a boundary line \(x - y = 1\). You would then determine which side of this line satisfies the inequality by testing points or looking at the graphical representation.

• Linear inequalities define regions in the coordinate plane.
• Solutions include all the points in the defined region.
• They are represented by shaded areas on a graph.

Understanding linear inequalities' graphical representation helps in solving and identifying solutions visually, crucial for understading systems of inequalities.