Problem 13 - Algebra for College Students

Question

\(-x+4 y-4 \leq 0\)

Step-by-Step Solution

Answer

The solution is the region below the line \(y = \frac{1}{4}x + 1\), including the line itself.

Step 1: Convert the Inequality to Slope-Intercept Form

First, we want to rearrange the given inequality \(-x + 4y - 4 \leq 0\) so that it resembles the slope-intercept form of a line, \(y = mx + b\). Add \(x\) to both sides to isolate terms involving \(y\): \[4y - 4 \leq x\]Next, add 4 to both sides:\[4y \leq x + 4\]Finally, divide every term by 4 to solve for \(y\): \[y \leq \frac{1}{4}x + 1\].

Step 2: Determine the Boundary Line

The inequality \(y \leq \frac{1}{4}x + 1\) implies we can consider the line \(y = \frac{1}{4}x + 1\) as a boundary.The slope (\(m\)) of the line is \(\frac{1}{4}\), and the y-intercept (\(b\)) is 1. Thus, the boundary line is a solid line because the original inequality is \(\leq\), indicating the region on or below this line.

Step 3: Graph the Inequality

To graph the inequality \(y \leq \frac{1}{4}x + 1\):1. Start at the y-intercept (0, 1) on the graph.2. Use the slope \(\frac{1}{4}\) to determine the next point by rising 1 unit and running 4 units to the right, so another point is (4, 2).3. Draw a solid line through the points since the inequality is \(\leq\).4. Shade the region below the line to represent all the points \((x, y)\) that satisfy the inequality \(y \leq \frac{1}{4}x + 1\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward and very useful way to express a linear equation. It is written as: \[ y = mx + b \] where:

• \( m \) represents the slope of the line, showing the change in the y-value for a one-unit increase in the x-value.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

In the exercise provided, you began with the inequality \(-x + 4y - 4 \leq 0\). To make it easier to graph, you rearrange it into slope-intercept form: \[ y \leq \frac{1}{4}x + 1 \].This transformation sets you up perfectly for graphing and clearly shows the slope (\( \frac{1}{4} \)) and the y-intercept (1). These two components are vital because they allow you to draw the line accurately on a graph.

Graphing Inequalities

Graphing inequalities involves more than just drawing a line. It also includes identifying the area of the graph that satisfies the inequality. Let's break this down:

• Start by graphing the boundary line. For the inequality \( y \leq \frac{1}{4}x + 1 \), use the slope (\( \frac{1}{4} \)) and the y-intercept (1) to sketch the line.
• Since the inequality sign is \( \leq \), you draw a solid line. If it were \( < \), a dashed line would be used instead to indicate that points on the line are not included in the solution.
• Next, determine which area of the graph fulfills the inequality. Since you have \( y \leq \), you shade below the line, capturing all the points (x, y) that meet the condition.

By following these steps, you ensure that all solutions to the inequality are represented on the graph.

Boundary Line

A boundary line is essential in graphing inequalities because it defines where one side of the inequality changes to the other. Here are the key details:

• For the given inequality \( y \leq \frac{1}{4}x + 1 \), the line \( y = \frac{1}{4}x + 1 \) acts as the boundary.
• The boundary line is drawn using the slope and y-intercept from the slope-intercept form.
• Since the inequality includes "equals" (\( \leq \) or \( \geq \)), the boundary is depicted as a solid line. This means that points on the line are also part of the solution set.

The role of the boundary line is to divide the graph into two regions—one that satisfies the inequality and one that does not. This helps you accurately depict all potential solutions on the graph.