Problem 63 - Algebra 2

Question

Graph each inequality. $$ y>\frac{2}{3} x-3 $$

Step-by-Step Solution

Answer

Graph the line \(y = \frac{2}{3}x - 3\), dashed, and shade above the line.

Step 1: Understanding the Inequality

The given inequality is \( y > \frac{2}{3}x - 3 \). This is a linear inequality, similar to a linear equation, but it uses a greater-than symbol, which means the region above the line will be shaded.

Step 2: Identify the Equation of the Boundary Line

Before graphing the inequality, we need to graph the boundary line, which has an equation like the inequality but using an equal sign. The equation is \( y = \frac{2}{3}x - 3 \).

Step 3: Graph the Boundary Line

1. Plot the y-intercept: The y-intercept is \(-3\). Mark this point on the graph at \( (0, -3) \). 2. Use the slope \(\frac{2}{3}\) to find another point: From the y-intercept \( (0,-3) \), move up 2 units and right 3 units to get the next point at \((3, -1)\). 3. Draw a dashed line through these points: Since the inequality is \(>\), the line is not included in the solution set, and should be dashed.

Step 4: Shade the Correct Region

Since the inequality is \( y > \frac{2}{3}x - 3 \), shade the region above the boundary line. You can pick a test point, like the origin \((0, 0)\), and check if it satisfies the inequality. If it does, shade the region containing the test point, otherwise, shade the opposite side. In this case, \(0 > \frac{2}{3}(0) - 3\) is true, so the origin is part of the solution set and you shade above the line.

Key Concepts

Linear Inequality

A linear inequality is similar to a linear equation but with a twist. Instead of having an "equals" sign, it might have an inequality sign like ">", "<", "≥", or "≤". This changes how we interpret the line on a graph.
Understanding linear inequalities is crucial as they represent a range of values instead of a single solution. This allows us to express a relationship where a variable can have many possible values that satisfy the condition set by the inequality.
Linear inequalities can describe constraints and conditions in real-world scenarios, such as budgeting, resource allocation, and optimizing solutions. Grasping their graphical representation will help in visualizing these scenarios.

Boundary Line

In graphing a linear inequality, the boundary line plays a pivotal role. It defines one of the edges of the possible solutions to the inequality.
The boundary line is formed by temporarily changing the inequality sign to an equal sign. For instance, for the inequality \( y > \frac{2}{3}x - 3 \), the boundary line would be \( y = \frac{2}{3}x - 3 \).
Once plotted, this line acts as a divider between two sets of possible solution areas on the graph.

• For the "greater than" (>) or "less than" (<) inequalities, the boundary line is drawn as a dashed line. It indicates that points on this line are not included in the solution set.
• For "greater than or equal to" (≥) or "less than or equal to" (≤) inequalities, use a solid line, since points on the line themselves are valid solutions.

Slope-Intercept Form

The slope-intercept form of a line is the most straightforward way to express a linear equation. This form is written as \( y = mx + b \), where \( m \) stands for the slope and \( b \) is the y-intercept.
The slope (\( m \)) indicates the tilt of the line, showing how much the line rises or falls as you move along the x-axis. For example, a slope of \( \frac{2}{3} \) signifies a rise of 2 units up for every 3 units across.
The y-intercept (\( b \)) is where the line crosses the y-axis. It tells us the value of \( y \) when \( x = 0 \). In our example, the line crosses at \( -3 \).
Understanding this form is essential for graphing, as it provides the direction and starting point for plotting the line.

Shading Regions

When graphing an inequality, shading illustrates all possible solutions that satisfy the inequality. After plotting the boundary line, the next step is determining which side of the line represents the solution set.
Shading the region involves:

• First, pick a test point not on the boundary line (often the origin, \( (0,0) \), if it's not on the line).
• Substitute this test point in the inequality. If the inequality holds true, then the region containing the test point is shaded.
• If the test point does not satisfy the inequality, shade the opposite side.

This shading visually shows where all the solutions to the inequality exist. It emphasizes the range of values that satisfy the condition, making it easier to understand which values work.