Problem 38 - Precalculus with Limits

Question

In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system of inequalities. $$\left\\{\begin{aligned} x-7 y>&-36 \\ 5 x+2 y>& 5 \\ 6 x-5 y>& 6 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution region for the given system of inequalities is represented by the shaded area on the plot, below the lines \(y=\frac{1}{7}x+5.14\), \(y=\frac{5-x}{2}\) and \(y=\frac{6-x}{5}\), where the three regions overlap.

Step 1: Derive the Equations from Inequalities

First, rewrite each inequality in the y=mx+b form (slope-intercept form) in order to plot them on a graph. The inequalities \(x-7 y>-36\), \(5 x+2 y>5\) and \(6 x-5 y>6\) become \(y<\frac{1}{7}x+5.14\), \(y<\frac{5-x}{2}\) and \(y<\frac{6-x}{5}\) respectively after rearranging.

Step 2: Plotting the Lines

Next, plot these lines on the graph. Remember that the inequalities are 'greater than' inequalities, so the lines will be dashed lines, indicating that the points on the lines are not included in the solution.

Step 3: Shading the Solution Region

Once all lines are plotted, determine the side of the line where the solutions lie for each inequality. For all three inequalities, the solution will be below the lines as each inequality is 'greater than'. Therefore, shade the region below the lines. The shared overlapped area among the three inequalities represents the solution region for this system of inequalities.

Key Concepts

Plotting Inequalities

Plotting inequalities on a graph is a visual way to represent the solution to a system of inequalities. To plot inequalities, you must first transform them into equations and then graph these equations as if they were lines. For example, the inequality \(x - 7y > -36\) can be rewritten in slope-intercept form, \(y < \frac{1}{7}x + 5.14\), which is essential for graphing.

Once you have the equation of a line, you plot it using a standard Cartesian plane, typically labeling the x and y-axes. However, unlike simple line equations, inequalities indicate a range of values for the y variable, so you have to remember to use a dashed line to show that the inequality does not include equality. Moreover, after plotting the dashed line, you need to shade the appropriate side of the line to display which side of the plane satisfies the inequality.

Slope-Intercept Form

The slope-intercept form of a line, \(y = mx + b\), is a formula that allows you to graph linear equations easily. In this form, \(m\) represents the slope of the line, which tells you how steep the line is, and \(b\) represents the y-intercept, which is where the line crosses the y-axis.

Using the slope-intercept form makes it easier to plot points and draw the line. You start by plotting the y-intercept, the point \((0, b)\), and then using the slope \(m\) as a ratio of rise over run to determine another point on the line. From the y-intercept, you go up if \(m\) is positive or down if it's negative, then move to the right to plot your second point, and then draw the line through both points.

Solution Region

In graphing systems of inequalities, the solution region is the area on the graph where all the inequalities in the system are satisfied simultaneously. After you plot all the inequalities, you'll notice that each one divides the plane into two halves and one half of each will be shaded.

Where the shaded areas overlap is your solution region. This is the set of all the points that satisfy all the inequalities at once. In the exercise provided, once all lines are plotted and the appropriate sides shaded, the solution region is the overlapped area below all three lines. This region could be bounded or unbounded, meaning it could be limited to a certain area or it could extend infinitely in one or more directions.

Dashed Lines in Inequalities

When graphing inequalities, it's important to differentiate between 'less than or equal to' and 'greater than or equal to' situations, which use solid lines, versus 'less than' and 'greater than' situations, which use dashed lines. The dashed lines indicate that the exact values on the line are not included in the solution set.

In the given exercise, all the lines are dashed because each original inequality uses a 'greater than' sign. It's crucial to represent this correctly: the dashed line serves as a boundary that the solution region approaches but never includes. When you shade the area for the inequality, remember not to shade the line itself, preserving the distinction that the points directly on the line do not satisfy the inequality.