Problem 34 - Precalculus with Limits

Question

In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system of inequalities. $$\left\\{\begin{array}{c}{3 x+4 y<12} \\ {x>0} \\ {y>0}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set is the region on the graph which is bounded by the axes and the line \(3x + 4y < 12\), and lies in the first quadrant.

Step 1: Drawing the Graph for Each Inequality

We firstly sketch each of the given inequalities separately on a coordinate plane.For \(3x + 4y < 12\), firstly, convert it to the standard form of a line equation \(y = mx + c\) by dividing everything by 4: \(y < -0.75x + 3\), it is a negative sloping line where y intercept is 3 and x intercept is 4. Because of the '<', the region below the line is the solution set for this inequality.For \(x > 0\), this inequality states that all the positive x values are in the solution set. It's a vertical line going through the origin. The solution set is to the right of this line.For \(y > 0\), similar to the previous inequality, this represents all positive y value. This is a horizontal line going through the origin. The solution set is above this line.

Step 2: Merging the Graphs

Next, merge all the graphs onto a single coordinate plane. As the first inequality has a negative slope, it descends from the y-intercept at y=3, crossing the x axis at x=4 and separates the plane into two parts. Based on '<', shade the part where the y-values are less than the x. The other two inequalities result in shading above the x-axis and to the right of the y-axis.

Step 3: Identifying the solution set

The solution to the system of inequalities is the region where the shadings from all the inequalities overlap. On the graph, it is the shaded region bounded by the axes and the line. We label the vertices as follows: origin (0,0), x-intercept (4,0), y-intercept (0,3) of the line \(3x + 4y < 12\).

Key Concepts

Sketching Inequality Graphs

Understanding how to sketch inequality graphs is essential to solving visual problems in algebra. When dealing with a system of inequalities, such as \(3x + 4y < 12\), \(x > 0\), and \(y > 0\), the goal is to find which points on the coordinate plane satisfy all of the given conditions simultaneously.

The first step is converting the inequality into an equation that represents a boundary line. If the inequality symbol is '<' or '>', the line will be dashed indicating that points on the line are not included in the solution set. If the inequality has the '\