Problem 8 - College Algebra

Question

In Exercises 1–26, graph each inequality. $$y>3 x+2$$

Step-by-Step Solution

Answer

To graph this inequality \(y > 3x + 2\), first plot the line \(y = 3x + 2\). Then identify the solution area to be above the line and shade it. The graphed line will be a dotted one since it's not included in the solution.

Step 1: Plot the Corresponding Equality

First, plot the line for the equation \(y = 3x + 2\). This is a straight line with a slope of 3 and y-intercept of 2.

Step 2: Identify the Boundary Line and Solution Area

The plotted line \(y = 3x + 2\) is the boundary of the solution. Since our inequality is \(y > 3x + 2\), the solution area is all above the line. The boundary line will not be included in the solution as it's just 'greater than (>)' not 'greater than or equal to (≥)'. The line will be drawn as a dotted one to represent this.

Step 3: Shade the Solution Area

The final step is to shade all the area that is above the line. This shading indicates that any point in this area is a solution to the given inequality.

Key Concepts

Slope-Intercept Form

When graphing linear inequalities, the slope-intercept form is incredibly useful. This form is written as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope tells you how steep the line is, or how much \(y\) changes for a change in \(x\).
For example, a slope of 3 means that for every 1 unit you move to the right on the x-axis, the y-value increases by 3 units. The y-intercept is where the line crosses the y-axis. It gives you a starting point on the graph when \(x = 0\).

• If the slope is positive, the line goes upwards as it moves from left to right.
• If the slope is negative, the line goes downwards as it moves from left to right.

In our inequality \(y > 3x + 2\), the slope is 3 and the y-intercept is 2, so the line will have a positive slope and will cross the y-axis at 2.

Boundary Line

The concept of the boundary line is essential when dealing with graphing inequalities. The boundary line is derived from the equation you get when switching an inequality sign for an equals sign. For \(y > 3x + 2\), the boundary line is \(y = 3x + 2\).
This line divides the plane into two distinct regions - one that satisfies the inequality and one that does not. For strict inequalities (\(>\) or \(<\)), as in our example, the line itself is not included in the solution set. We represent this on a graph with a dotted or dashed line.

• A solid line is used for \(≥\) or \(≤\) inequalities since points on the line satisfy the condition.
• A dotted line indicates that points on this line do not satisfy the inequality, which is why it's used here.

The boundary line acts as a visual guide for where the solution area starts.

Shading Solution Area

Shading is the final step in graphing linear inequalities and helps illustrate the solution area—the set of all points that satisfy the inequality. In the inequality \(y > 3x + 2\), the solution area is above the boundary line because \(y\) values must be greater than the corresponding \(y\) values on the line.
To determine the shading direction, you can use a test point. A common test point is the origin \((0,0)\), as it is easy to calculate. Substitute it into the inequality to see if it makes the inequality true.

• If \((0,0)\) satisfies the inequality, you shade the region containing this point.
• If it doesn't, you shade the opposite region.

In our case, substituting \(0\) for both \(x\) and \(y\) gives \(0 > 2\), which is false, so we shade the area not containing \((0,0)\), which is above the line. This shading visually represents all solutions to the inequality.