Problem 1 - Elementary Algebra Within Reach

Question

In Exercises \(1-6\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y \geq 3 x-3 \\ y \leq-x+1 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph of the system of inequalities consists of two lines representing the equations \(y = 3x - 3\) and \(y = -x + 1\). The solution area is the intersecting region of the shaded areas above the line for \(y \geq 3x - 3\) and below the line for \(y \leq -x + 1\).

Step 1: Identifying the equations

Identify the equations of the lines in the system. Here, these are \(y = 3x - 3\) and \(y = -x + 1\).

Step 2: Drawing the graphs

For the equation \(y = 3x -3\), it is a straight line with a slope of 3 and cuts the y-axis at -3. Sketch this line. Similarly sketch the line for \(y = -x + 1\), which has a slope of -1 and cuts the y-axis at 1. These lines divide the graph into several regions.

Step 3: Shading the area for \(y \geq 3x - 3\)

Choose a test point that's not on the line \(y = 3x - 3\), say (0, 0). Substituting these coordinates into the inequality, \(0 \geq 3(0) - 3\), gives false. This means the solution area to the inequality is the area not containing the point (0, 0), i.e., the area above the line. Shade this area.

Step 4: Shading the area for \(y \leq -x + 1\)

Similarly, for the inequality \(y \leq -x + 1\), choose a test point not on the line, say (0,0). Substituting into the inequality, \(0 \leq -0 + 1\), gives true. This means the solution to this inequality is the area containing the point (0, 0), i.e., the area below the line. Shade this area.

Step 5: The solution area

The solution to the system of inequalities is represented by the intersecting region of the shaded areas from step 3 and step 4.

Key Concepts

linear equations

Linear equations are the backbone of the graph you are about to draw. They form straight lines when graphed on the coordinate plane. A linear equation in two variables is typically formatted as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.A vital aspect of understanding linear equations is recognizing that they essentially depict a consistent relationship between two quantities. For example, in the equation \( y = 3x - 3 \), the slope \( m \) of 3 tells us that for every unit increase in \( x \), \( y \) increases by 3 units.These equations allow us to create a visual representation of a balanced relationship, giving a reference point for more complex mathematical concepts like inequalities.

slope-intercept form

The slope-intercept form is one of the most common ways to express linear equations. It is written as \( y = mx + b \), where \( m \) represents the slope, and \( b \) represents the y-intercept.

• The slope \( m \) shows how steep the line is. A positive slope indicates the line rises from left to right, while a negative slope means it falls.
• The y-intercept \( b \) is where the line intersects the y-axis. For example, in \( y = -x + 1 \), the line crosses the y-axis at 1.

Using this form simplifies the process of graphing because it provides direct information about a line's trajectory and starting point on the graph.By converting any linear equation into this form, you can easily identify its properties and sketch it with confidence.

systems of inequalities

A system of inequalities is a collection of two or more inequalities linked by the common variables. These systems describe a range of solutions rather than a single point, showcasing an area on the graph where the solution set lies. In our exercise, you see inequalities like:

• \( y \geq 3x - 3 \)
• \( y \leq -x + 1 \)

When you deal with a system such as this, the goal is to find the intersection of all viable solutions, which represents the set of points satisfying every inequality.Think of it as overlap areas on the graph where each condition holds. By cleverly choosing test points and shading the areas accordingly, you can delineate where these overlaps occur.This shaded region is the graphical solution to the system, helping visualize how different conditions interact on a coordinate plane.

graphing techniques

Graphing techniques for inequalities involve a few extra steps compared to graphing linear equations. Here’s how you can effectively approach this:First, plot the related linear equations. This is done by marking the y-intercept and using the slope to determine another point. For example, with \( y = 3x - 3 \), start at (0, -3) and move up 3, right 1 repeatedly, drawing the line.Next, determine the type of boundary by checking the inequality symbol:

• \( \geq \) or \( \leq \) use solid lines to indicate the points on the line are included in the solution.
• Strict inequalities \( > \) or \( < \) use dashed lines to show the points are not part of the solution.

Use a test point, such as (0, 0), to see which side of the line should be shaded. If the test point satisfies the inequality, shade that side of the line.Finally, locate the solution region for the system by finding where the shaded areas intersect. This organized approach helps clarify complex inequalities through simple, visual steps on a graph.