Problem 27 - College Algebra with Modeling and Visualization

Question

Graph the solution set to the system of inequalities. $$ \begin{aligned} &x-2 y \geq 0\\\ &x-3 y \leq 3 \end{aligned} $$

Step-by-Step Solution

Answer

The solution set is the overlapping region, where one is below \(y = \frac{1}{2}x\) and above \(y = \frac{1}{3}x - 1\).

Step 1: Write Each Inequality in Slope-Intercept Form

To graph each inequality, it's helpful to write them in the slope-intercept form, which is \(y = mx + b\). - For \(x - 2y \geq 0\), rearrange it as \(2y \leq x\) or \(y \leq \frac{1}{2}x\).- For \(x - 3y \leq 3\), rearrange it as \(3y \geq x - 3\) or \(y \geq \frac{1}{3}x - 1\). These forms help in plotting the boundary lines and determining the regions.

Step 2: Graph the Boundary Lines

Graph the boundary lines for each inequality. Convert each inequality to an equation by replacing the inequality symbol with an equal sign. - Line for \(y = \frac{1}{2}x\) (a solid line, since the inequality is \(\leq\)).- Line for \(y = \frac{1}{3}x - 1\) (a solid line, since the inequality is \(\geq\)).Plot these lines on the coordinate plane.

Step 3: Determine the Shading for Each Inequality

Shade the appropriate side of each line based on the inequality:- For \(y \leq \frac{1}{2}x\), shade below the line \(y = \frac{1}{2}x\). This represents all the points \((x, y)\) that satisfy the inequality.- For \(y \geq \frac{1}{3}x - 1\), shade above the line \(y = \frac{1}{3}x - 1\). This represents all the points \((x, y)\) that satisfy the inequality.

Step 4: Identify the Solution Region

The solution region is where the shaded areas from both inequalities overlap. This is the set of points \((x, y)\) that satisfy both inequalities at the same time.

Key Concepts

Slope-Intercept Form

To tackle systems of linear inequalities, a great starting point is to convert each inequality into slope-intercept form. This form looks like \(y = mx + b\), with \(m\) representing the slope and \(b\) the y-intercept.
This format is advantageous because it easily connects with how lines are plotted on a graph.

• Rewriting Inequality 1: Start with \(x - 2y \geq 0\). By rearranging terms, we get \(2y \leq x\). Dividing every term by 2 gives us \(y \leq \frac{1}{2}x\).
• Rewriting Inequality 2: For \(x - 3y \leq 3\), rearrange to \(3y \geq x - 3\). Dividing by 3 leads to \(y \geq \frac{1}{3}x - 1\).

Once in this form, the equations become easy to visualize and plot.

Graphing Inequalities

Graphing inequalities involves a few steps after converting them into slope-intercept form. You need to plot boundary lines and decide where to shade on the graph.
Start by graphing the boundary lines which replace the inequality with an equal sign.

• Line for Inequality 1: For \(y = \frac{1}{2}x\), plot this as a solid line. A solid line means the line itself is part of the solution since the inequality symbol is inclusive (\( \leq\)).
• Line for Inequality 2: Plot \(y = \frac{1}{3}x - 1\) as a solid line for the same reason, relating to the \( \geq\) symbol.

After plotting these lines, the next step involves shading. For the first inequality, shade below the line, since it’s \( \leq\). For the second, shade above the line, considering \( \geq\).
This shading indicates which region satisfies each inequality.

Solution Region

The ultimate goal when graphing a system of inequalities is to find the solution region. This is where the shaded regions of each individual inequality intersect or overlap on the graph.

• Overlap Identification: Once both inequalities are graphed and shaded, look for the area where both shadings meet. This overlap is the portion of the graph that satisfies every inequality in the system.
• Shared Solutions: The coordinates within this overlapping area meet the conditions specified by all the inequalities. Hence, they are part of the solution set.

This region is the answer to the system of inequalities, effectively visualizing which \((x, y)\) pairs work for all equations involved. It's like finding a middle ground where all conditions agree, making it a key component of solving such systems.