Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}x-y &<1 \\\\-1

Step-by-Step Solution

Answer

Graph the overlapping region above \(y = x - 1\), above \(y = -1\), and below \(y = 1\).

Step 1: Understand Each Inequality

The system of inequalities we need to solve is: 1) \(x - y < 1\) and 2) \(-1 < y < 1\). Each inequality represents a region on the coordinate plane. We need to graph these inequalities to find the overlapping region.

Step 2: Graph the First Inequality

For the inequality \(x - y < 1\), rewrite it in slope-intercept form as \(y > x - 1\). This inequality means that we graph the line \(y = x - 1\) as a dashed line (because the inequality is strict and does not include the boundary), and shade above the line because \(y\) is greater than \(x - 1\).

Step 3: Graph the Second Inequality Part 1

Handle \(-1 < y\) by treating it as an inequality separate from \(y < 1\). Graph the line \(y = -1\) as a dashed line and shade the area above this line, since \(y\) is greater than \(-1\).

Step 4: Graph the Second Inequality Part 2

Now handle \(y < 1\). Graph the line \(y = 1\) also as a dashed line and shade the area below this line, since \(y\) is less than \(1\).

Step 5: Determine the Overlapping Region

The solution set of the system consists of the region that satisfies all inequalities. So, the overlap of the shaded area above the line \(y = x - 1\), above \(y = -1\), and below \(y = 1\) is the desired solution region.

Key Concepts

Graphing Inequalities

Graphing inequalities involves creating a visual representation of the solutions to the inequality on a coordinate plane. Each inequality results in a specific area, illustrating all the points that satisfy the inequality. When graphing, you:

• Convert the inequality into an equation to find the boundary line.
• Draw the boundary line on the graph. For a strict inequality (like "<" or ">"), use a dashed line to show that points on the line are not included in the solution.
• For inequalities "≤" or "≥", draw a solid line instead, as points on this line do belong to the solution set.
• Shade the appropriate side of the boundary line to represent all solutions to the inequality. This side changes based on the direction of the inequality symbol.

The overall goal is to clearly convey the range of values that make the inequality true, thus helping locate the solution set of the system.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers known as coordinates. These coordinates are written as \(x, y\), where "x" represents the horizontal axis (also known as the x-axis), and "y" represents the vertical axis (also known as the y-axis).

To better understand the inequalities, we need to plot them on the coordinate plane:

• Identify the x and y axes, and mark units equally along these axes.
• Plot any relevant points according to their \(x, y\) values.
• Use these points to draw the lines that act as boundaries for each inequality.

Using the coordinate plane, you can visualize how different inequalities interact with each other, crucial for finding regions that satisfy multiple inequalities.

Solution Set

The solution set of a system of inequalities is the collection of all points (ordered pairs) that simultaneously satisfy all considered inequalities. When graphing, it’s crucial to:

• Observe where shading from different inequalities overlap on the graph.
• Only regions where all inequalities overlap are considered part of the solution set.
• The solution set can be infinite and is usually a region or area contained within the coordinate plane.

For a practical example with the given equations:

The solution set comprises the area above the line \(y = x - 1\), above \(y = -1\), and below \(y = 1\).

This understanding helps determine where the values meet the conditions set by all parts of the system.

Overlapping Region

The overlapping region is where the shaded parts from all the graph's inequalities intersect. This area represents the solution set as it includes all the points that satisfy every inequality in the system.

• Begin by identifying the shaded area for each inequality individually on the graph.
• Examine where these shaded areas meet. The common area marks the overlapping region.
• In our example, the overlap lies between three boundaries: above \(y = x - 1\), above \(y = -1\), and below \(y = 1\).

The overlapping region is vital in solving systems of inequalities since it visually demonstrates the set of solutions that satisfy all conditions of the problem.