Problem 55 - Thinking Mathematically

Question

Explain how to graph \(2 x-3 y<6\).

Step-by-Step Solution

Answer

In summary, to graph the inequality \(2x - 3y < 6\), first, rearrange the inequality into slope-intercept form, then graph the line as a dashed line. Finally, shade the region above the line.

Step 1: Convert the inequality into the slope-intercept form

The slope-intercept form is \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. To do this, you'll need to rearrange the inequality as follows: Subtract \(2x\) from both sides to get \(-3y < -2x + 6\). Then, divide every term by \(-3\). However, keep in mind that when you divide or multiply an inequality by a negative number, the inequality sign flips. So, the inequality becomes \(y > 2/3x -2\)

Step 2: Graph the line \(y = 2/3x - 2\)

First, plot the y-intercept \(-2\) on the y-axis. Then, use the slope to find another point on the graph. The slope is \(2/3\), meaning for every 3 units you move to the right from the y-intercept, move 2 units up. Then, draw a line through these points. Since the inequality symbol is '>', this will be a dashed line.

Step 3: Shade the correct area

The inequality is \(y > 2/3x - 2\). This tells us that the solution contains points where y-values are greater than those on the line. So, you'll shade the area above the line.

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for graphing linear equations and inequalities. It is expressed as \( y = mx + c \) , where \( m \) stands for the slope of the line, and \( c \) is the y-intercept, the point where the line crosses the y-axis. To graph an inequality like \( 2x-3y < 6 \) , we start by isolating \( y \) . Subtracting \( 2x \) from both sides gives us \( -3y < -2x + 6 \) . When we divide by \( -3 \) to solve for \( y \) , we get the inequality in the slope-intercept form \( y > \frac{2}{3}x - 2 \) . This form tells us exactly how to plot the line and proceed with the graphing process.

Inequality Signs Flipping

When dealing with inequalities, the direction of the inequality sign is of utmost importance. A common point of confusion arises when we multiply or divide both sides of an inequality by a negative number which causes the inequality sign to flip. For example, if we have \( -3y < -2x + 6 \) and we divide by \( -3 \) , the '<' becomes '>'. This is a crucial step to remember because it changes the direction of the inequality and affects which side of the line will be included in the solution set. Always check the sign when manipulating inequalities to ensure accurate solutions.

Graphing Inequalities

Graphing inequalities involves a few more steps than graphing equalities. Once the inequality is in slope-intercept form, like \( y > \frac{2}{3}x - 2 \) , plot the y-intercept on the y-axis. Here, it's \( -2 \) . Next, use the slope \( \frac{2}{3} \) to find another point. From the y-intercept, move right 3 units and up 2 units, to follow the rise-over-run rule. With these points, draw a dashed line—dashed because the inequality is strict ( '>' and not '\textgreater=')—which indicates that points on the line itself are not part of the solution. The line is a border that separates the graph into two regions, one that satisfies the inequality and one that does not.

Shading Solution Regions

The final step in graphing inequalities is to identify the solution region, which is represented by shading on the graph. The direction of the inequality sign tells us where to shade. For \( y > \frac{2}{3}x - 2 \) , where \( y \) is greater than the expression, we shade above the line. The shaded area represents all the points where the inequality holds true. It's the graphical representation of the solution set and helps to visually convey the range of possible answers to the inequality. Shading correctly is essential—it's the difference between the right solution and a common mistake, especially when graphing solutions to inequalities.