Problem 115 - College Algebra

Question

Sketch the graph of the solution set for the following system of inequalities: $$\left\\{\begin{array}{l} y \geq n x+b(n<0, b>0) \\ y \leq m x+b(m>0, b>0). \end{array}\right.$$

Step-by-Step Solution

Answer

The graph of the system of inequalities is a vertical striped region between two lines starting at \((0,b)\) with the first one having a negative slope and the second one a positive slope.

Step 1: Sketch the graph for the first inequality

The first inequality is \(y \geq n x+b\), it is a linear inequality, so it is represented by a line in the plane. Given \(n<0\), this is a decreasing slope line and above this line since \(y\) is greater or equal. The \(y\)-intercept is the constant term \(b\) which is positive. Therefore, the line starts from \((0,b)\) and goes downwards as \(x\) increases.

Step 2: Sketch the graph for the second inequality

The next inequality is \(y \leq m x+b\), which will create another line. As \(m>0\), the line will have an increasing slope and since \(y\) is less than or equal to this line, the region below the line is considered. The \(y\)-intercept is also \(b\), therefore, this line also starts from \((0,b)\) and goes upwards as \(x\) increases.

Step 3: Find the solution set for the system of inequalities

The intersection of the regions determined by each of the lines from step 1 and step 2 represents the solution for the system of inequalities. These regions are formed above the line from the first inequality and under the line from the second one. The graph for the system of inequalities is a vertical striped area between the two lines starting at \((0,b)\).

Key Concepts

Graphical Representation of Systems of Inequalities

Visualizing a system of inequalities can greatly aid in understanding their solution set. A system might consist of two or more inequalities that are combined, and the solution is where the conditions of all these inequalities are fulfilled simultaneously.
For a given system of inequalities, each inequality is represented by its respective line on a coordinate plane:

• Lines visible in the plane mark the boundary where inequalities equal, such as when the inequality symbols are replaced with equality.
• The inequalities themselves indicate which side of these lines contains the solutions.

By shading or highlighting the region representing each inequality, the graphical solution set becomes apparent where these regions overlap. This intersection of shaded areas illustrates the feasible combinations of values, satisfying all provided conditions.

Understanding Linear Inequalities

Linear inequalities resemble linear equations but include inequality symbols such as \(\geq\) or \(\leq\) instead of an equals sign. These symbols indicate a range of values allowed for solutions, rather than just a specific point.
In the context of a linear inequality:

• The inequality can either suggest a range that includes the boundary line (using symbols \(\leq\) or \(\geq\)) or exclude it (using \(<\) or \(>\)).
• Each linear inequality typically relates two variables, and when represented graphically, forms a dividing line on a plane.

Understanding how to interpret these dividing lines is essential, as they show whether a solution should reside above, below, or on the line itself, which directly influences the solution set of the inequalities together.

Defining the Solution Set

The solution set for a system of inequalities comprises all possible values that satisfy every condition of the system simultaneously. It is not a single point, but often a region on a graph divided by one or more lines resulting from each inequality.
For instance:

• With our system, the solution set is depicted as the space where the shaded regions of both inequalities overlap.
• These regions illustrate the comprehensive set of \((x,y)\) pairs that fit within both constraints.

In graphical terms, the solution set is often represented as a space "trapped" by the boundary lines of inequalities, showcasing all the potential solutions collectively.

Intersection of Regions in Inequalities

Intersection of regions is key in identifying where all solutions lie in tandem for a set of inequalities. Each inequality's graphical representation forms a separate region on the plane:

• One inequality might carve out a region above its boundary line.
• The other could form a region below its respective line.

The intersection, or overlap, of these areas marks the complete solution set where both conditions are met. Just like in our exercise, you might have a visually distinct, shaded region indicating where solutions exist for the whole system.