Problem 55 - Algebra and Trigonometry

Question

Shading Unwanted Regions To graph the solution of a system of inequalities, we have shaded the solution of each inequality in a different color; the solution of the system is the region where all the shaded parts overlap. Here is a different method: For each inequality, shade the region that does not satisfy the inequality. Explain why the part of the plane that is left unshaded is the solution of the system. Solve the following system by both methods. Which do you prefer? Why? $$ \left\\{\begin{aligned} x+2 y &>4 \\\\-x+y &<1 \\ x+3 y &<9 \\ x &<3 \end{aligned}\right. $$

Step-by-Step Solution

Answer

Both methods find the solution region; shading unwanted regions gives a clearer solution.

Step 1: Graph the first inequality

First, solve the first inequality for the boundary line by setting it to an equation: \(x + 2y = 4\). Rewriting it in slope-intercept form gives \(y = -\frac{1}{2}x + 2\). Draw this line on the graph. Since it is a 'greater than' inequality, shade the region above the line to show the solutions of \(x + 2y > 4\).

Step 2: Graph the second inequality

Set the second inequality equal to find the boundary line: \(-x + y = 1\). This equals \(y = x + 1\). Draw this line on the graph. Since it's a 'less than' inequality, shade the region below the line to represent \(-x + y < 1\).

Step 3: Graph the third inequality

For the third inequality \(x + 3y < 9\), the boundary line would be \(x + 3y = 9\), or \(y = -\frac{1}{3}x + 3\). Plot this line and shade below it as the inequality is a 'less than'. This shows the solutions to \(x + 3y < 9\).

Step 4: Graph the fourth inequality

The fourth inequality is \(x < 3\). For this, draw a vertical line at \(x = 3\) and shade to the left of this line because it represents the region where \(x < 3\).

Step 5: Find the overlapping region

The solution to the system of inequalities is the region where all the shaded areas from the first three sums meet. However, using the original method, we shade the opposite for each inequality and find the unshaded area is the solution: it is where none of the 'not-satisfying' regions overlap.

Step 6: Shading using the opposite method

Now, shade the opposite regions for each inequality: for \(x + 2y > 4\), shade below; for \(-x + y < 1\), shade above; for \(x + 3y < 9\), shade above; and for \(x < 3\), shade right. The non-overlapped and unshaded region is the solution to the system.

Step 7: Conclusion: Preference of methods

Both methods yield the same solution region, but personally, using the 'shading unwanted regions' method is preferable as it might result in less clutter on the graph and visually highlights the solution area more clearly.

Key Concepts

Graphing Inequalities

Graphing inequalities is the process of visually representing a range of solutions on a coordinate plane. Unlike equalities, which are represented by a line, inequalities introduce a region of solutions. To graph an inequality:

• Begin by manipulating the inequality into a linear equation format to easily graph the boundary line.
• Identify whether the inequality is 'greater than' or 'less than' to determine the proper shading direction.
• Use a dotted line to indicate inequalities that don't include equality (>, <) and a solid line if they do (≥, ≤).

This approach helps to visualize the infinite possibilities that satisfy an inequality.

Solution Region

The solution region is the specific part of the graph that satisfies all inequalities in a system. It’s essentially the real-world "answer" to the question, where all conditions are met simultaneously. To find this region:

• Graph each inequality on the same coordinate plane.
• Identify where the shading overlaps for each inequality; this intersection is the "solution region".
• This region represents every possible coordinate pair (x, y) that satisfies all provided inequalities.

In practical terms, this space might represent potential solutions to a business problem or feasible scenarios in a physics experiment.

Shading Method

The shading method is a crucial step in graphing systems of inequalities. It helps in clearly identifying which parts of the graph do or do not satisfy the inequality equations:

• For each inequality, decide which side of the boundary line represents solutions that satisfy the inequality.
• Shade this side to indicate those points meet the inequality condition.
• Sometimes, it's beneficial to shade the opposite or unwanted region to easily spot the correct solution region when using multiple inequalities.

This method not only clarifies the solution visually but also aids in confirming that solutions meet all necessary conditions.

Graphical Representation

Graphical representation of systems of inequalities involves plotting all relevant lines and shaded regions on the same set of axes. This offers a visual overview of potential solutions:

• Start by plotting each boundary line by transforming each inequality into its equality form.
• Use distinct shading for each inequality for clarity.
• The solution to the system of inequalities is where all shaded regions overlap or, conversely, where none of the unwanted regions appear using the opposite shading method.

This graph not only simplifies complex algebraic systems but also aids in visual learning, making it easier to understand abstract mathematical concepts.