91Ó°ÊÓ

A system of inequalities consists of multiple inequalities that are considered simultaneously. In this exercise, we dealt with two inequalities:

• \(x + y < 4\)
• \(4x - 2y < 6\)

These inequalities define a range of values for which both conditions hold true at the same time.
The solution to this system is not just a single value but a region on the graph where both inequalities overlap. An important aspect of solving these systems is analyzing both equations together, rather than in isolation.
When considering a system of inequalities on a graph, regions are shaded to represent the range of solutions that satisfy each inequality. The area where these shaded regions overlap is the key to finding the common solutions.

The slope-intercept form of a linear equation is one of the most useful formats for graphing a line. It is given by the equation: \[y = mx + b\]where:

• \(m\) is the slope of the line, representing the steepness of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

In our exercise, we converted the inequalities into this form as follows:

• From \(x + y < 4\) to \(y < 4 - x\)
• From \(4x - 2y < 6\) to \(y > 3 - 2x\)

Converting to slope-intercept form makes it easier to graph each inequality. We are able to clearly distinguish the slope and y-intercept, facilitating an accurate representation of each line on a graph.

The Cartesian plane is a two-dimensional plane defined by an x-axis and a y-axis. Each point on this plane can be represented by a pair of numerical coordinates: \((x, y)\).
In graphing inequalities, the Cartesian plane provides a visual way to display all possible solutions. The lines corresponding to the equations are drawn based on their slope and y-intercept.

• A line separates the plane into two regions: one that satisfies the inequality and one that does not.
• Shading indicates the region where the inequality holds true.

In our example, both lines were drawn on the Cartesian plane, and the areas representing solutions to each inequality were identified by shading. This makes it straightforward to see where the solutions intersect.

The solution set is the collection of all possible solutions for a given system of inequalities. It is graphically represented as the overlap of shaded areas on the Cartesian plane.
For the system in the exercise, the solution set is the area that meets the conditions of both inequalities:

• Below the line \(y = 4 - x\)
• Above the line \(y = 3 - 2x\)

The region that satisfies both conditions is shaded, indicating that points in this region solve both inequalities simultaneously. Visualizing the solution set helps in understanding complex algebraic solutions and is crucial for interpreting the results of the system of inequalities.