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Solving a system of inequalities graphically involves plotting lines on a coordinate plane that represent each inequality in the system. We start by rewriting the inequalities in slope-intercept form:

• For example, the inequality \(x - y \leq 0\) becomes \(y \geq x\).
• Similarly, \(2x + 3y \geq 10\) becomes \(y \geq -\frac{2}{3}x + \frac{10}{3}\).

Having the inequalities in slope-intercept form simplifies the process because this form is easy to graph. The slope-intercept form \(y = mx + b\) shows you the slope \(m\) of the line and the y-intercept \(b\) where the line crosses the y-axis.
For each inequality, we draw a boundary line on a graph. If an inequality is "greater than or equal to" or "less than or equal to," the boundary line is solid, indicating that points on the line satisfy the inequality.
After graphing each line, you determine the feasible region by shading the side of the line where the inequality holds true. The overlapping shaded region represents the set of solutions to the system of inequalities.

Once you graph the inequalities, observe the shaded area that represents possible solutions. This region is known as the feasible region. Depending on the boundaries created by the inequalities, the feasible region can be

• Bounded: Enclosed on all sides by lines or curves, forming a closed shape such as a triangle or polygon. Such regions contain all solutions within a finite area.
• Unbounded: Open on at least one side, extending infinitely in that direction. An unbounded region means the solutions extend infinitely and the feasible region does not form a closed shape.

In our example, we noticed the shaded area from the inequalities extends infinitely upwards and to the right, meaning there are no boundaries enclosing the feasible region completely. Therefore, the solution set is unbounded, indicating the set of solutions is infinite in extent.

The slope-intercept form, \(y = mx + b\), is an equation format where a straight line is described by two main components: the slope \(m\) and the y-intercept \(b\). Understanding this form is crucial for graphing and solving inequalities. Here’s what each component represents:

• The slope \(m\) indicates how steep the line is. It tells us how much \(y\) changes for a unit change in \(x\). If \(m\) is positive, the line rises as \(x\) increases; if negative, the line falls as \(x\) increases.
• The y-intercept \(b\) shows where the line crosses the y-axis. This starting point helps in sketching the initial portion of the line on the graph.

To tackle inequalities, convert each inequality into the slope-intercept form. This enables you to easily plot the line and decide where the shaded regions belong. For example, in our exercise, converting \(2x + 3y \geq 10\) into \(y \geq -\frac{2}{3}x + \frac{10}{3}\) enables a straightforward graphing process, showcasing both the slope and the y-intercept for clarity.