Problem 9 - Thinking Mathematically

Question

Graph each linear inequality. \(2 y-3 x>6\)

Step-by-Step Solution

Answer

The inequality \(y > 1.5x + 3\) is graphed as a dashed line. Because the test point (0,0) does not satisfy the inequality, the area above the line is shaded.

Step 1: Convert the inequality to slope-intercept form

Isolate \(y\) on one side of the inequality. Subtract -3x from both sides to produce \(2y > 3x + 6\). Now, divide every term by 2 to isolate \(y\). This leads to \(y > 1.5x + 3\)

Step 2: Graph the inequality

Initially, graph the line \(y = 1.5x + 3\), making it a dashed line, because the inequality is '>' and not '\u2265'. Next, identify a test point not on the line, usually (0,0), and substitute these coordinates into the original inequality. If the inequality holds, shade the side of the line where the test point lies, else, shade the opposite side.

Step 3: Test the inequality

Use the point (0,0) for this test. Substitute these values into the original inequality \(2y -3x > 6\). Since this test gives \(0 > 6\), which is false, shade the side of the line that doesn't contain the origin (0,0).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when graphing linear inequalities. In its simplest terms, this form is given by the equation \( y = mx + b \), where

• \( m \) represents the slope of the line, which indicates how steep the line is, and
• \( b \) represents the y-intercept, which is the point at which the line crosses the y-axis.

To graph an inequality like \( 2y - 3x > 6 \), following the step-by-step solution, we first convert it into slope-intercept form. Here, we solve for \( y \) to get \( y > 1.5x + 3 \). The number 1.5 represents our slope \( m \), and 3 is our y-intercept \( b \). This shows the initial line we need to graph, which is foundational, as the inequality will determine which side of this line is relevant to the solution.

Inequalities in Algebra

Dealing with inequalities in algebra may seem daunting, but they follow logic similar to equations with a twist. Inequalities don't have just one solution but rather a range of possible solutions.
When graphing inequalities, such as \( 2y - 3x > 6 \), we are looking not just for a line, but for a region that satisfies the inequality. The inequality sign '>' tells us that the region we are interested in is above the line \( y = 1.5x + 3 \). But, unlike equations, where we draw a solid line, inequalities require us to consider boundary lines that might or might not be part of the solution.
The dashed line in this case indicates that the points on the line are not included in the solution set because our inequality strictly states 'greater than' and not 'greater than or equal to'. Remember to look at the type of inequality sign to decide whether to use a solid or dashed line.

Test Point Method

The test point method is a simple yet powerful technique to determine which side of the boundary line contains the solution to an inequality. After graphing the dashed or solid line:

• Pick a test point not on the boundary line.
• Substitute the coordinates of this point into the original inequality.
• Determine whether the inequality remains true with these values.

In the solution provided, (0,0) was used as a test point. When plugged into the inequality \( 2y - 3x > 6 \), it resulted in \( 0 > 6 \), which is false, indicating that the origin is not part of the solution set. Therefore, we shade the region of the graph that does not include the origin.
Most often (0,0) is used for its simplicity, but if the boundary line passes through the origin, another point should be chosen. Utilizing the test point method is a quick way to visually communicate the solutions to an inequality on a graph.