Problem 32 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}2 x-6 y>12 \\\x \leq 0 \\\y \leq 0\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the shaded triangular region in the 3rd Quadrant of the Cartesian Plane.

Step 1: Plotting Inequalities

First is to plot the inequalities on the Cartesian plane. The line for the first inequality 2x - 6y > 12 can be simplified into y < 1/3x - 2 and will be represented by a dashed line since it is greater than. Interchanging the signs, we get y > 1/3x -2, which corresponds to the half plane above the line. The other inequalities 'x ≤ 0' and 'y≤ 0' will get solid lines since they're 'less than or equal to'. The inequality x ≤ 0 corresponds to all points that are on or to the left of the y-axis, and similarly y≤ 0 corresponds to all points that are on or below the x-axis.

Step 2: Finding the Intersection Area

The solution set is the intersection of all three half-planes represented by the inequalities. To find that, we shade the regions representing each inequality on the Cartesian plane and look for a common intersection area which would be the answer to the system of inequalities.

Step 3: Graphing the Solution Set

The solution set is the shaded region satisfying all three inequalities. This region represents all the (x,y) pairs that solve the given system of inequalities. In this case, it is the triangular region formed by the three lines in the 3rd Quadrant.

Key Concepts

Cartesian Plane

The Cartesian plane, also known as the coordinate plane, is the foundation for graphing linear inequalities.
It consists of two axes: the horizontal x-axis and the vertical y-axis. These axes intersect at a point called the origin, denoted as (0, 0).
The plane divides into four quadrants, which help to determine the location of all points (x, y).

• Quadrant I: Both x and y values are positive.
• Quadrant II: x is negative and y is positive.
• Quadrant III: Both x and y values are negative.
• Quadrant IV: x is positive and y is negative.

In the context of linear inequalities, the Cartesian plane serves as the canvas where we can visualize the range of solutions.
Each inequality defines a region on this plane, and understanding which quadrant the region lies in is crucial to solving systems of inequalities.

Inequality Graphing

Graphing inequalities involves drawing boundary lines on the Cartesian plane to define areas that satisfy each inequality.
For the inequality system being solved, **2x - 6y > 12** becomes **y < \(\frac{1}{3}x - 2\)** when simplified, indicating a dashed line.
This type of line shows that points on the line are not part of the solution set.

• Dashed lines represent strict inequalities, like > or <.
• Solid lines are used for ≤ or ≥ to include the boundary in the solution set.

For the second inequality **x ≤ 0**, the line is solid, marking all points on and to the left of the y-axis.
The third inequality **y ≤ 0** is also solid, covering all points on and below the x-axis.
Effectively graphing these inequalities is crucial and helps visualize their intersections.

Solution Set of Inequalities

The solution set of a system of inequalities is the region where all specified conditions are satisfied simultaneously.
In this exercise, the solution involves identifying the overlap between regions defined by each of the inequalities.
This intersection is critical, as it represents the set of all possible (x, y) values that satisfy every inequality in the system.
For example:

• The inequality y < \(\frac{1}{3}x - 2\) suggests the solution set lies below the line.
• The conditions x ≤ 0 and y ≤ 0 restricts the solution set to the third quadrant.

The region where these zones overlap becomes the solution set.
In this case, the area is triangular and falls within the third quadrant, visually showing the complete solution space. Obtaining this solution requires careful analysis and plotting of each inequality involved.