Problem 51 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&3 x+y \leq 6\\\&2 x-y \leq-1\\\&x \geq-2\\\&y \leq 4\end{aligned} $$

Step-by-Step Solution

Answer

The system of inequalities has a solution. The solution is the region where all four shaded areas from each inequality overlap on the graph.

Step 1: Graph each inequality separately

Start by graphing each inequality separately. This can be done by first graphing the line that would correspond to each if they were equations (i.e. replace the inequality sign with an equal sign) and then shading the appropriate area. So, each inequality would correspond to the lines \(3x + y = 6\), \(2x - y = -1\), \(x = -2\), \(y = 4\).

Step 2: Visualize which side of the area to shade

After drawing the lines, determine which side of the area to shade for each inequality. Remember, with inequalities shaded area denotes the solution to the inequality. For \(3x + y \leq 6\) and \(y \leq 4\), shade the area below both lines. For \(2x - y \leq -1\), shade the area above the line. And for \(x \geq -2\), shade the area to the right of the line.

Step 3: Identify the region where all conditions are satisfied

Then, the final solution to the system of inequalities is the area where the shading from all four inequalities overlaps, that is, where all conditions are satisfied simultaneously. This is the intersection of all the separate solutions.

Step 4: Confirm if there is a solution

If there is no region where all four shading areas overlap, then the system has no solution. But in this case, there is an intersection of all the shaded areas implying the system of inequalities has a solution

Key Concepts

Graphing Inequalities

Graphing inequalities involves converting an inequality into a graphical form by treating the inequality as an equation. This helps to visually decipher which parts of the coordinate plane satisfy the inequality. Start by graphing the lines as if the inequality signs are equal signs. For instance:

• For the inequality \(3x + y \leq 6\), graph the line \(3x + y = 6\).
• For \(2x - y \leq -1\), graph \(2x - y = -1\).
• The lines \(x = -2\) and \(y = 4\) are vertical and horizontal lines respectively.

Each line represents a boundary on the graph and initially divides the plane into two regions. These regions will either satisfy the inequality or not. Proper line work lays the foundation for finding solutions by shading the right side.

Solution Set

The solution set of a system of inequalities is the collection of all points that satisfy all of the inequalities simultaneously. When you graph inequalities, the solution set manifests as a region on the graph. This region is found by:

• Identifying the correct side to shade for each inequality.
• Finding where the shaded areas overlap.

In this particular exercise, the solution set isn't just a random area. It's where all logical parameters dictated by the inequalities come together: all equations need to hold true at the same time. Therefore, the points within the solution set meet every single condition given by each inequality in the system.

Shading Regions

Shading is a visual method used to show the solution set on a graph. Once the boundary lines are plotted, the next step is to determine which side of the line to shade.When shading:

• For inequalities using "\(\leq\)" or "\(\geq\)", shade the region including the boundary line, which indicates the points on the line are part of the solution set.
• For \(3x + y \leq 6\), you'd shade below the line \(3x + y = 6\).
• If an inequality is strictly less than or greater than, you would not include the line itself in the shading.

Differentiate each shading distinctly to avoid confusion, often using different shades or patterns. This way, you can easily identify the overlap in the next step.

Intersection of Inequalities

The intersection of inequalities is where all the shaded regions converge on the graph. This intersection is the heart of the solution set. Finding the intersection involves:

• Looking for where all the individual shaded areas overlap.
• Ensuring that every condition from each inequality is satisfied.

For this exercise, the overlap between the shaded portions for all inequalities identifies to the solution of the system. The intersection confirms whether there is a viable solution set or not. If no overlap exists, it signals that there is no solution meeting all the inequality conditions.