Problem 28 - 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 to the system of inequalities is the overlapping region between the two shaded areas from step 1 and 2.

Step 1: Graph the first inequality

First, graph the inequality \(3x+6y \leq 6\). To do this, we first change the inequality into the form \(y \leq mx+b\), where m is the slope and b is the y-intercept. By dividing throughout by 6, we get \(y \leq -0.5x + 1\). This tells us that the line intersects with the y-axis at y=1 and has a slope of -0.5. Plot the line and shade the region including the line and to the bottom of it.

Step 2: Graph the second inequality

For the second inequality \(2x + y \leq 8\), change it to the form \(y \leq mx + b\) by subtracting 2x from both sides. This gives us \(y \leq -2x + 8\). This tells us that the line intersects with the y-axis at y=8 and has a slope of -2. Plot the line and shade the region including the line and to the bottom of it.

Step 3: Find the overlapping region

The solution to the system of inequalities is the region where the shaded areas from step 1 and 2 overlap. This is the set of points that satisfy both inequalities.