Problem 72 - College Algebra

Question

The graphs of solution sets of systems of inequalities involve finding the intersection of the solution sets of two or more inequalities. By contrast, in Exercises \(71-72,\) you will be graphing the union of the solution sets of two inequalities. Graph the union of \(x-y \geq-1\) and \(5 x-2 y \leq 10\).

Step-by-Step Solution

Answer

The solution is any point that lies in either the area defined by the inequality \(y \leq x + 1\) or the area defined by the inequality \(y \geq \frac{5}{2}x - 5\).

Step 1: Graph the First Inequality

Start by graphing the first inequality \(x-y \geq -1\). This can be rewritten in slope-intercept form as \(y \leq x + 1\). This results in a straight line with slope 1 and y-intercept -1. Since the inequality is 'less than or equal to', all points on the line and all points below the line are part of the solution set.

Step 2: Graph the Second Inequality

Similarly, for the second inequality \(5x-2y \leq 10\), which can be rewritten in slope-intercept form as \(y \geq \frac{5}{2}x - 5\). This results in a straight line with slope \(5/2\) and y-intercept \(-5\). As the inequality is 'greater than or equal to', all points on the line and above the line are solutions.

Step 3: Find the Union of Solution Sets

Since the task is to find the union of the solution sets, after graphing, shading the areas dictated by both inequalities, the solution is the combination of both areas. Therefore, any point that lies in either of the two shaded regions is the solution.

Key Concepts

Solution Sets of Inequalities

Understanding the solution sets of inequalities is crucial when dealing with graphing systems of inequalities. A solution set represents all the possible solutions that satisfy a given inequality. When graphing a single linear inequality, such as \( y \textless x + 3 \), you shade the region of the graph where all points satisfy this condition.

For the inequality \( x-y \textgreater= -1 \), the solution set includes all the points that lie on or above the line \( y = x + 1 \) because the inequality indicates 'greater than or equal to.' In the case of inequalities, the lines can be solid (for 'greater than or equal to' or 'less than or equal to') or dashed ('greater than' or 'less than'). The solution set for a single inequality is either above or below the line, depending on the direction of the inequality.

When working with multiple inequalities, understanding each individual solution set is the first step before finding their intersection or union.

Union of Inequalities

The union of solution sets of multiple inequalities is another key concept in graphing systems of inequalities. When instructed to find the union, you are looking for any point that satisfies at least one of the inequalities involved.

In the exercise provided, you must first graph the two inequalities \( x-y \textgreater= -1 \) and \( 5x-2y \textless= 10 \). After graphing each line and shading their respective solution sets, you combine these regions. The union represents all the points that lie in either of the shaded areas or where the areas overlap.

Visualization of Union

In this graphing exercise, pay attention to how the shaded areas from each inequality intersect and where they do not. The overall solution set for the union will look like a combination of both individual sets. This can sometimes result in a larger region compared to when you're finding the intersection of solution sets, which represents points that satisfy all the inequalities.

Slope-Intercept Form

The slope-intercept form of a line is an essential tool for graphing systems of inequalities. This form is written as \( y = mx + b \), where \( m \) is the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis.

In the initial step of our example solution, the first inequality \( x-y \textgreater= -1 \) is rearranged into slope-intercept form as \( y \textless= x + 1 \). This tells us directly that the line has a slope of 1 (it rises one unit for every unit it moves to the right) and crosses the y-axis at -1. Similarly, the second inequality can be expressed in this form as \( y \textgreater= (5/2)x - 5 \), indicating a steeper slope of 5/2 and a y-intercept of -5.

The ease of graphing these inequalities once they are in slope-intercept form cannot be overstated. It provides a clear starting point and direction for drawing the line. Moreover, it simplifies the process of identifying whether to shade above or below the line, as dictated by the inequality sign in the original equation.