Problem 12 - Algebra for College Students

Question

For Problems \(1-24\), indicate the solution set for each system of inequalities by graphing the system and shading the appropriate region. $$ \left(\begin{array}{l} y \leq x+2 \\ y \geq x \end{array}\right) $$

Step-by-Step Solution

Answer

The solution set is the region between the lines \( y = x \) and \( y = x + 2 \), inclusive of both lines.

Step 1: Graph the First Inequality

The first inequality is \( y \leq x + 2 \). To graph this, draw the line \( y = x + 2 \), which is a line with a slope of 1 and a y-intercept at (0, 2). Since the inequality is "less than or equal to," shade the region below the line, including the line itself.

Step 2: Graph the Second Inequality

The second inequality is \( y \geq x \). Graph the line \( y = x \), which is a line with a slope of 1 and a y-intercept at (0, 0). Since the inequality is "greater than or equal to," shade the region above the line, including the line itself.

Step 3: Identify the Solution Set

The solution set is the region where the shaded areas from both inequalities overlap. This region will include the line \( y = x \) and continue below the line \( y = x + 2 \). Points in this overlap satisfy both inequalities.

Key Concepts

Graphing Inequalities

When we talk about graphing inequalities, we're discussing a way to find all possible solutions that satisfy a given condition. Imagine you're setting up a boundary, and your job is to find out which side of the line contains the solutions. Graphing inequalities means plotting the boundary as a line on a graph and then figuring out which region on the graph meets the conditions stated by the inequality.

• Start by transforming the inequality into an equation. For example, with the inequality \( y \leq x + 2 \), the boundary line would be \( y = x + 2 \). This line acts as your guide.
• Next, look at the inequality symbol to decide where to shade. For \( y \leq x + 2 \), you will shade below the line because you're looking for values of \( y \) that are less than or equal to what you have on the line.
• Use solid lines for inequalities that include "less than or equal to" (\( \leq \)) or "greater than or equal to" (\( \geq \)), showing that the line itself is part of the solution set.

By understanding these steps, you can graph any linear inequality and find its solution region.

Solution Set

A solution set for a system of inequalities is simply the collection of points that satisfy all the inequalities in the system. Imagine throwing a huge net across a graph and finding exactly where all the conditions overlap.
When graphing multiple inequalities, each condition will have its own shaded region. The solution set appears where these shaded areas intersect.

• For our system of inequalities \( y \leq x+2 \) and \( y \geq x \), we graphed each by shading the appropriate regions. The solution set is the overlap of these regions.
• The overlap region is where both conditions meet. Here, you could test a point, like \( (0, 0) \), within this overlap to ensure it's part of the solution set.
• This region tells us that any point within it satisfies both inequalities simultaneously.

Finding the solution set is all about locating that perfect intersection, where all conditions are satisfied at once.

Linear Equations

Linear equations are the cornerstone of graphing and solving systems of inequalities. They represent straight lines on a graph and form the boundaries for our inequalities.
A linear equation is expressed generally as \( y = mx + b \), where:

• \( m \) is the slope, telling you how steep the line is.
• \( b \) is the y-intercept, showing where the line crosses the y-axis.

In our system, the equations \( y = x + 2 \) and \( y = x \) serve as the boundary lines for graphing our inequalities.

• \( y = x + 2 \) has a slope of 1 and meets the y-axis at (0, 2).
• \( y = x \) also has a slope of 1 but intercepts at (0, 0).

These lines help us define the regions of solutions, which is particularly important when defining the constraints of a solution set. Understanding how to graph and position these lines helps accurately illustrate all potential solutions for the given system.