Problem 8 - A Graphical Approach to College Algebra

Question

Graph each inequality. Do not use a calculator. $$x<3+2 y$$

Step-by-Step Solution

Answer

Graph the line from (3, 0) to (5, 1) as a dashed line, and shade the region below the line.

Step 1: Convert Inequality to Equation

The first step to graphing an inequality is to convert it into an equation, as the line will represent the boundary. So, take the inequality \(x < 3 + 2y\) and convert it into the equation \(x = 3 + 2y\).

Step 2: Rearrange the Equation

We rearrange the equation to better understand the structure of the line. \(x = 3 + 2y\) can also be written in terms of \(y\):\[ y = \frac{x - 3}{2} \] This is in slope-intercept form \(y = mx + c\).

Step 3: Determine Points for the Line

Choose values for \(x\) to find corresponding \(y\) values. Suppose \(x = 3\), then \(y = 0\). If \(x = 5\), substituting gives: \(y = 1\) because:\[ y = \frac{5 - 3}{2} = 1 \]. Use these points (3, 0) and (5, 1) to plot the line on a graph.

Step 4: Draw the Boundary Line

Use the points found in the previous step to draw the boundary line on a graph. Since the inequality is \(x < 3 + 2y\), we draw a dashed line to indicate that points on the line are not included within the solution set.

Step 5: Shade the Solution Region

Since the inequality is \(x < 3 + 2y\), we need to determine which side of the line includes solutions to the inequality. Choose a test point not on the line, such as \((0, 0)\). Substitute into the inequality: \(0 < 3 + 2(0)\) gives \(0 < 3\), which is true, so the region towards the origin should be shaded.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a powerful tool in graphing equations and understanding linear relationships. It is typically expressed as \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) denotes the y-intercept. This form makes it easy to identify how a line behaves and its position on a coordinate plane.

• Slope (\( m \)): Indicates the steepness of the line, showing how much \( y \) changes as \( x \) increases by one unit. A positive slope means the line rises, while a negative slope means it falls.
• Y-intercept (\( c \)): Reveals the point where the line crosses the y-axis. Understanding this helps place the line accurately on the graph.

Converting an equation to slope-intercept form makes plotting straightforward, revealing relationships between variables instantly.

Boundary Line

The boundary line is a crucial element when dealing with inequalities. It represents the line that separates possible solutions.

• To find the boundary line, you first convert the inequality into an equation.
• This line can be solid or dashed. In our example, we use a dashed line because the inequality is less than (<), meaning points on the line aren't included in the solution set.

Visualizing the boundary helps identify regions that satisfy given inequalities and effectively partition the graph.

Solution Region

The solution region is where all the answers to an inequality lie. It’s the part of the graph that either includes or excludes the boundary line.

• After drawing the boundary line, you determine which side contains solutions.
• This is done by shading the region that satisfies the inequality, often based on a test point. In our example, the area towards the origin satisfied the condition \( x < 3 + 2y \).

Finding this area helps in visualizing and understanding the inequality's constraints more concretely.

Test Point Method

The test point method is a straightforward way to determine which side of the boundary line satisfies the inequality.

• Start by choosing a test point that isn’t on the boundary line to avoid errors. A common choice is the origin, \((0, 0)\), if it isn't on the line.
• Substitute this point into the inequality. If it makes the inequality true, the side containing this point is the solution region.
• If it's false, the opposite side of the line holds the solutions.

This method simplifies the process of identifying the correct solution region, ensuring accuracy in graphing and interpreting inequalities.