Problem 21 - Precalculus

Question

In Exercises \(21-50,\) graph each system of inequalities or indicate that the system has no solution. $$\begin{aligned} &y \geq x-1\\\ &y \leq x+1 \end{aligned}$$

Step-by-Step Solution

Answer

The solution is the region between the lines \(y = x - 1\) and \(y = x + 1\), including both lines.

Step 1: Graph the Line of the First Inequality

The first inequality given is \( y \geq x - 1 \). Start by converting the inequality to the equation \( y = x - 1 \). Plot two points using this equation, for example, if \( x = 0 \), then \( y = -1 \), and if \( x = 2 \), then \( y = 1 \). Plot these points and draw a solid line through them on the coordinate plane because the inequality includes \(y = x - 1\). Shade the region above the line to indicate \( y \geq x - 1 \).

Step 2: Graph the Line of the Second Inequality

The second inequality is \( y \leq x + 1 \). Similarly, start by converting it to \( y = x + 1 \). Choose two points; for example, if \( x = 0 \), then \( y = 1 \), and if \( x = 2 \), then \( y = 3 \). Plot these points and draw a solid line through them as well, since the inequality includes \( y = x + 1 \). Shade the region below this line to represent \( y \leq x + 1 \).

Step 3: Identify the Solution of the System of Inequalities

Now observe the shaded regions of both inequalities on the graph. The solution for the system of inequalities is the area where the shaded regions intersect. Look for the region that is both above the line \( y = x - 1 \) and below the line \( y = x + 1 \). This overlapping shaded area represents the solution to the system of inequalities.

Key Concepts

Graphing Inequalities

When working with systems of inequalities, a good starting point is to individually graph each inequality. The key is transforming each inequality into a standard equation form, such as turning the inequality \( y \geq x-1 \) into \( y = x-1 \).
First, plot this line on a coordinate graph. Since the inequality sign includes \( \geq \), the line itself is a part of the solution, so it should be drawn as a solid line.

• Pick two easy points that satisfy the equation, such as \( (0, -1) \) and \( (2, 1) \).
• Connect these points with a straight line.

Once your line is in place, it represents a boundary between solutions and non-solutions. For systems with more than one inequality, repeat this process for the next inequality as well, ensuring accurate representation of each line on the graph.

Shading Regions

Shading is a crucial step when solving inequalities on a graph, as it visually outlines the solutions. To shade regions, remember:

• The shading direction tells you where the value of \( y \) satisfies the inequality in relation to the line.
• For \( y \geq x-1 \), shade above the line, as it indicates \( y \) values are more than or equal to \( x-1 \).
• For \( y \leq x+1 \), shade below the line because the \( y \) values are less than or equal to \( x+1 \).

Shading not only highlights the set of solutions but also helps to visualize where multiple inequalities might overlap and intersect. Darkening or cross-hatching often makes the region clearer where these inequalities simultaneously hold true.

Intersection of Inequalities

The intersection of inequalities refers to the solution set where the shaded parts of each inequality on the graph overlap. This region represents the values of \( x \) and \( y \) that satisfy all the inequalities in the system.
For the example provided, you will see two overlapping shaded areas:

• A top section, shaded for \( y \geq x-1 \).
• A bottom section, shaded for \( y \leq x+1 \).

Identifying this common area is crucial as it answers the system of inequalities. It’s the real-world solution where both conditions (or more, if there are more inequalities) are valid. This area can either be a clear intersection or sometimes, there might be no intersection at all, implying no solution exists.