Problem 95 - College Algebra

Question

Explain how to graph the solution set of a system of inequalities.

Step-by-Step Solution

Answer

The graph of the system of inequalities typically involves graphing each inequality on the same set of axes. One would shade above the line for inequalities showing y is greater, or beneath the line when y is less. The solution set where these shadings overlap is the pairs of \(x, y\) that satisfy both inequalities simultaneously.

Step 1: Identify the Inequalities

In order to begin, you need to know the inequalities being dealt with. Let's consider a system of inequalities, for example, \(y \geq x + 1\) and \(y < 3x - 1\). These will be graphed on the same coordinate system.

Step 2: Graph the First Inequality

The first step is to graph the first inequality, \(y \geq x + 1\), as if it were a linear equation. Plot the line \(y = x + 1\). Since the inequality symbol is '\(\geq\)', this requires a solid line because the line is included in the solution set. Then, because \(y\) is greater or equal to \(x + 1\), the area above the line is shaded.

Step 3: Graph the Second Inequality

Next graph the second inequality, \(y < 3x - 1\), like a linear equation. Plot the line \(y = 3x - 1\). Since the inequality symbol is '<', a dashed line is required because the line is not included in the solution set. Since \(y\) is less than \(3x - 1\), the area below the line is shaded.

Step 4: Identify the Solution Set

The solution set is the area where the shadings from both inequalities overlap. This area represents all the pairs of \(x, y\) that satisfy both inequalities simultaneously.