Problem 4 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{c}4 x+3 y \leq 12 \\\x-2 y \leq 4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set of the system of inequalities \(4x + 3y \leq 12\) and \(x - 2y \leq 4\) can be represented by the intersection of the two shaded regions when the inequalities are graphed.

Step 1: Graphing the first inequality

To graph \(4x + 3y \leq 12\), first turn it into an equation \(4x + 3y = 12\) and plot this line on the cartesian grid. Lines for inequalities are drawn solid if the inequality is \(\leq\) or \(\geq\) and dashed if \(<\) or \(>\). Here it will be solid because the inequality is \(\leq\). After graphing the line, choose a test point not on the line (usually (0,0) if it is not on the line) and substitute its coordinates into the inequality. If the inequality is true, shade the region containing the test point, otherwise, shade the opposite region.

Step 2: Graphing the second inequality

To graph \(x - 2y \leq 4\), same process as above is followed. First turn it into an equation \(x - 2y = 4\) and plot this line. As the inequality is \(\leq\), the line will be solid. After graphing the line, choose a test point not on the line and substitute its coordinates into the inequality. If it satisfies the inequality, shade the region containing the test point, otherwise, shade the opposite region.

Step 3: Find the intersection of the two inequalities

The intersection of the two shaded regions from step 1 and step 2 represents the solution sets of the system of inequalities. It contains all the points satifying both inequalities at the same time.

Key Concepts

Solution Set

When dealing with a system of inequalities, you're actually searching for the solution set, which includes all the points that satisfy both inequalities simultaneously. This is different from the traditional solution to an equation, which might be a single point or a few points.

For the given system of inequalities:

• \(4x + 3y \leq 12\)
• \(x - 2y \leq 4\)

We aim to find the region where both conditions are true. This involves graphing each inequality on the same coordinate plane and determining where their regions overlap.

Think of it like trying to find a common area on a Venn diagram where two groups overlap. Any point in this overlapping area is a solution to both inequalities.

System of Inequalities

A system of inequalities is a set of two or more inequalities with the same variables. Working with them involves figuring out where all the conditions can be met at once.

In our example, the system includes two inequalities:

• \(4x + 3y \leq 12\)
• \(x - 2y \leq 4\)

Each inequality represents a half-plane on the graph. By graphing them, you can visually see the areas that satisfy each inequality. The intersection of these areas gives you the solution to the system.

So, always remember:

• Convert each inequality to an equation to find the boundary line.
• Determine which side of the line the inequality represents.
• Look for the area where the solutions to all inequalities overlap.

Shading Regions

Shading regions is a visual way to show where the solutions to an inequality lie on a graph. Here's how it works:

After drawing the line for an inequality, decide which side to shade by using a test point. A common choice for the test point is \((0,0)\) unless it lies on the line.

• If substituting the test point into the inequality gives a true statement, shade the area containing the test point.
• If it’s false, shade the opposite side.

For the inequality \(4x + 3y \leq 12\), if \((0,0)\) satisfies the inequality, you shade inside. Otherwise, you shade outside.

In our system, after shading for both inequalities, the overlapping shaded area indicates the solution set. This shaded intersection effectively pinpoints all the possible solutions that fit both inequalities.