Problem 49 - 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? $$\left\\{\begin{aligned} x+2 y &>4 \\\\-x+y &<1 \\ x+3 y &<9 \\ x &<3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Both methods lead to the same solution. Some may find the traditional method more intuitive.

Step 1: Rearrange Inequalities

Convert each inequality into an equation to plot the boundary lines. 1. From \(x + 2y > 4\), rearrange to \(2y = -x + 4\). The boundary line is \(2y = -x + 4\) or \(y = -\frac{1}{2}x + 2\).2. From \(-x + y < 1\), rearrange to \(y = x + 1\).3. From \(x + 3y < 9\), rearrange to \(3y = -x + 9\) or \(y = -\frac{1}{3}x + 3\).4. From \(x < 3\), the boundary line is \(x = 3\).

Step 2: Plot Boundary Lines

Graph each boundary line on the coordinate plane using dashed lines, because the inequalities are strict (>, <) rather than \(\geq\) or \(\leq\).

Step 3: Traditional Method - Shade Satisfying Regions

For each inequality, determine which side of the line satisfies the inequality by testing points:- \(x + 2y > 4\) means shading above the line \(y = -\frac{1}{2}x + 2\).- \(-x + y < 1\) implies shading below the line \(y = x + 1\).- \(x + 3y < 9\) means shading below the line \(y = -\frac{1}{3}x + 3\).- \(x < 3\) means shading left of the vertical line \(x = 3\).Find the region where all the shaded areas overlap, which represents the solution.

Step 4: Alternative Method - Shade Non-Satisfying Regions

This approach involves shading the opposite regions:- For \(x + 2y > 4\), shade below the line \(y = -\frac{1}{2}x + 2\).- For \(-x + y < 1\), shade above the line \(y = x + 1\).- For \(x + 3y < 9\), shade above the line \(y = -\frac{1}{3}x + 3\).- For \(x < 3\), shade right of the line \(x = 3\).The non-shaded region after all inequalities are applied is the solution set.

Step 5: Compare Methods

Both methods lead to the same solution region, but some may prefer the traditional method as it directly shows where the solution is by overlap. The alternative method uses shading to eliminate non-solutions, potentially clarifying different visual strategies.

Key Concepts

Graphing Inequalities

Graphing inequalities effectively represents solutions on a graph. Unlike equations, which map onto precise lines or curves, inequalities show regions of possible solutions.

To graph an inequality, you first convert it into an equation to identify its boundary line. This line divides the plane into two sections, one being the solution region, the other not.

Remember to use dashed lines for inequalities that are "less than" or "greater than" ( > , <) because these do not include the boundary line. Solid lines are used for "less than or equal to" or "greater than or equal to" (≤, ≥), where the boundary line is part of the solution.

As you graph multiple inequalities, your job is to identify where these solution regions overlap, forming the complete solution to the system.

Solution of Inequalities

The solution of a system of inequalities is the region where all conditions set by the inequalities are satisfied simultaneously.

Solving inequalities graphically involves two common methods:

• Traditional Shading: Shade the area of the graph that meets the inequality condition.
• Alternative Shading: Shade the area that does not satisfy the condition. The region left unshaded after this method is the solution.

Both methods will pinpoint the same solution region, although the visualization process is different.

This dual approach can be quite helpful for students to choose the method they find easier to understand and apply.

No matter the method, confirming the results by checking test points is always an excellent way to ensure accuracy in identifying the solution region.

Boundary Lines

Boundary lines are key concepts when graphing inequalities; they serve as dividers on the graph between solution and non-solution regions.

These lines originate from rewriting inequalities as equations. For example, in the inequality \(x + 2y > 4\), the boundary line is interpreted as \(y = -\frac{1}{2}x + 2\). This line: indicates where the equality part ends and the inequality part begins.

It's important to plot boundary lines accurately, using dashed lines for strict inequalities and solid lines for inclusive ones.

Once both inequality and boundary lines are on the graph, testing a point on each side of the line helps determine which side reflects the true solution region.

Inequality Regions

Inequality regions refer to areas on the graph designated as part of the solution when solving inequalities.

For each inequality, the plane is divided by its boundary line, creating two main regions. One side satisfies the inequality, and the other does not.

When you have a system of inequalities, these regions interact. You look for overlap between the shaded regions in the traditional method or the remaining unshaded region in the alternative shading method.

To determine the inequality region, a common practice involves selecting test points in different sections of the graph. This testing confirms the inclusion or exclusion of regions for each inequality individually.

In doing so, the solution region is comprehensively and visually identified, ensuring clarity and accuracy in results.