Problem 27 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}3 x+6 y \leq 6 \\\2 x+y \leq 8\end{array}\right.$$

Step-by-Step Solution

Answer

The solution of the system of inequalities is the region below both lines \(y = 1 - x/2\) and \(y = 8 - 2x\), including the lines themselves. This region represents all points (x,y) such that \(x/2 + y \leq 1\) and \(2x + y \leq 8\).

Step 1: Transform inequalities to slope-intercept form

First, rearrange the inequalities so they're in slope-intercept form \(y=mx+b\). For the first inequality, divide by 6, yielding \(x/2 + y \leq 1\), or \(y \leq 1 - x/2\) . Similarly, for the second inequality, solve for \(y\), yielding \(y \leq 8 - 2x\).

Step 2: Graph the lines

Next, graph the lines \(y = 1 - x/2\) and \(y = 8 - 2x\) on a graph. Because the inequalities are \(\leq\), the lines themselves are included in the solution, so they should be solid lines.

Step 3: Determine the solution region

The solution of each inequality is the region below the line. The solution of the system is the region below both lines. Plot a test point, e.g., the origin (0,0), which should satisfy all inequalities. \(0 \leq 1\) and \(0 \leq 8\), so the origin is part of the solution region. Shade the area below both lines and modify the shading such that it only covers the region below both lines.

Key Concepts

Graphing Inequalities

Graphing inequalities helps us visualize the solution set of a system of inequalities. Start by understanding that a linear inequality divides the coordinate plane into two regions. One region represents solutions that satisfy the inequality, while the other does not.

To graph an inequality:

• Convert the inequality to its corresponding equation to plot the boundary line.
• Decide if the line is solid or dashed. Use a solid line for \(\leq\) or \(\geq\), indicating the boundary is inclusive. A dashed line is for \(<\) or \(>\), where it's not part of the solution.
• Choose a test point, often the origin (0,0), to determine which side of the line to shade. Substitute the point into the inequality:

If the inequality holds true for the test point, shade the region containing the point. If not, shade the opposite region. Repeat these steps for all inequalities to find the overall solution region.

Slope-Intercept Form

Understanding the slope-intercept form is essential for graphing linear relationships. This form is given by:

• \(y = mx + b\)
• Where \(m\) is the slope, describing the steepness and direction of the line.
• \(b\) represents the y-intercept, the point where the line crosses the y-axis.

For the inequality \(3x + 6y \leq 6\), convert to slope-intercept form by solving for \(y\). Divide every term by the coefficient of \(y\), resulting in \(y \leq 1 - x/2\). Similarly, transform \(2x + y \leq 8\) to \(y \leq 8 - 2x\). These modifications make it easier to graph each line and understand how they'll be positioned on the coordinate plane. Recognizing these key features helps in efficiently identifying and plotting the correct solution region.

Solution Region

The solution region of a system of inequalities is the overlap where all conditions are satisfied. After graphing each inequality, identify this common area by considering each inequality's individual solution region.

Use these steps to find it:

• Shade each inequality's solution set according to its graph.
• Find the intersection of all shaded areas.
• The overlap or common region represents the solution for the entire system. Any point within this area will satisfy all inequalities simultaneously.

For the given inequalities, \(y \leq 1 - x/2\) and \(y \leq 8 - 2x\), both solutions are below their respective lines. By shading both areas and noting their overlap, you define the solution region. Test points like the origin (0,0) can be used to confirm if they lie within this shared space. If your selected test point satisfies all inequalities, it belongs to the solution region.

Linear Inequalities

Linear inequalities are like linear equations but with inequality signs \(\leq, \geq, <, >\). They describe a range of possible solutions rather than a single line.

• Graphing turns them into dividing lines with areas of solution sets.
• Use a solid line for inclusive inequalities \((\geq, \leq)\) and a dashed line for non-inclusive \((<, >)\).

A significant aspect is separating the plane into regions, one being the solution region. Consider the direction of the inequality symbol and choose test points to verify which side of the boundary line belongs to the solution.

For the inequalities \(3x + 6y \leq 6\) and \(2x + y \leq 8\), the solutions are not a single point but all the points on or below both lines. Thus, a system of linear inequalities narrows down the infinite possibilities into a specific section of the graph. Understanding this visualization, we gain insight into how solutions interact and change with varying parameters.