Problem 49 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&x \geq 0\\\&y \geq 0\\\&2 x+5 y \leq 10\\\&3 x+4 y \leq 12\end{aligned} $$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the overlapped in all shaded areas from inequalities \(x \geq 0\), \(y \geq 0\), \(2x+5y \leq 10\) and \(3x+4y \leq 12\).

Step 1: Graph Inequality 1 and 2

Start the solution by drawing the Cartesian coordinate system. Then graph the first two inequalities \(x \geq 0\) and \(y \geq 0\). This is done by drawing a vertical line at \(x=0\) and a horizontal line at \(y=0\). Shade the area to the right of the vertical line (for \(x \geq 0\)) and above the horizontal line (for \(y \geq 0\)).

Step 2: Graph Inequality 3 and 4

Next, graph the third and fourth inequalities, \(2x+5y \leq 10\), and \(3x+4y \leq 12\). Start by setting each inequality to equal to create the lines corresponding to each inequality, \(2x+5y = 10\) and \(3x+4y = 12\). Graph those lines on the same coordinate system. For the inequality \(2x+5y \leq 10\), the line is solid (because of the \('less than or equal to'\) condition) and shading should be below the line. Same for the inequality \(3x+4y \leq 12\), shading should be below the line.

Step 3: Find the Intersection

Since the system of inequalities requires satisfying all four inequalities, find the intersection of shaded areas from Step 1 and Step 2. The overlapping shaded area in all four inequalities represents the solution set for the system of inequalities.

Key Concepts

Understanding the Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane defined by a pair of axes: the horizontal axis (x-axis) and the vertical axis (y-axis). These axes intersect at a point called the origin, which has coordinates (0,0). This system allows us to locate and graph points, lines, and shapes by using pairs of numbers.
The coordinate system divides the plane into four quadrants:

• Quadrant I: where both x and y are positive
• Quadrant II: where x is negative and y is positive
• Quadrant III: where both x and y are negative
• Quadrant IV: where x is positive and y is negative

When graphing inequalities, the Cartesian coordinate system helps visualize regions defined by these inequalities. Lines on the graph show borders where the inequality changes from true to false, such as in the inequalities involving x and y. By shading certain areas, we represent the range of solutions for those inequalities.

Defining the Solution Set

A solution set includes all possible values of variables that satisfy a given inequality or set of inequalities.
In the context of graphing, the solution set is the region where all the shaded areas from individual inequalities overlap.
For example, consider the inequalities given in the exercise:

• \(x \geq 0\) and \(y \geq 0\) describe the first quadrant, where x and y are non-negative.
• \(2x+5y \leq 10\): Graph this as a line and shade below it, since we want the values less than or equal to the line.
• \(3x+4y \leq 12\): Again, graph and shade below the line.

The overlap of these shaded regions is the feasible solution set for the system of inequalities. In this case, it's a polygonal area in the first quadrant of the coordinate plane.

Exploring a System of Inequalities

A system of inequalities consists of multiple inequalities that must be satisfied simultaneously. Each inequality describes a region in the Cartesian plane, and the solution to the system is the area where these regions intersect.
To solve a system of inequalities, follow these steps:

• Graph each inequality on the same coordinate system.
• Determine the region that satisfies each inequality, using shading to represent these areas.
• Identify the overlapping region where all conditions are satisfied. This overlap is the solution to the system.

In our exercise, the system includes two boundary conditions \(x \geq 0\) and \(y \geq 0\), ensuring solutions are within the first quadrant, and two linear inequalities, \(2x+5y \leq 10\) and \(3x+4y \leq 12\), each forming a line.
By following through these steps, we visually capture the solution set in the graph.