Problem 41 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}y \leq-3 x+3 \\\y \geq-x-1 \\\y

Step-by-Step Solution

Answer

The solution of the given system of linear inequalities will be the region on the graph where the shaded areas from all three inequalities intersect. If no such area exists, it means the system has no solutions.

Step 1: Graph the first inequality

Start by graphing the line \(y = -3x +3\). This line is the boundary for the inequality \(y \leq -3x +3\). Because of the 'less than or equal to' symbol, the shaded area will be below the line.

Step 2: Graph the second inequality

Next, draw the line for \(y = -x -1\). This line represents the boundary for the inequality \(y \geq -x -1\). Since it's a 'greater than or equal to' inequality, the area above this line will be shaded.

Step 3: Graph the third inequality

Finally, plot the line for \(y = x +7\). The inequality \(y < x + 7\) is represented by this line. It's a 'less than' inequality, meaning the shaded region will be below this line.

Step 4: Identify the solution set

The solution set of the system of inequalities is where the shaded areas of all three inequalities intersect on the graph. That will be the region that satisfies all three inequalities.

Step 5: No solution case

If there's no region that satisfies all three inequalities on the graph, the system has no solutions. This would occur if the shaded regions from the inequalities never intersect.

Key Concepts

Inequality Graphing

Inequality graphing is a visual method to represent solutions to inequalities on a coordinate plane. Imagine a simple equation like \( y = 2x + 1 \), which forms a straight line. An inequality such as \( y > 2x + 1 \) takes this a step further, indicating not just a line, but all the points above the line where the inequality holds true.

When graphing inequalities, we use a dashed line to represent a 'less than' or 'greater than' boundary (the points on the line are not included), and a solid line for 'less than or equal to' or 'greater than or equal to' (including the points on the line). The region of the graph that represents the solution is then shaded. For instance, if we have \( y \< x + 7 \), we'd graph a dashed line for \( y = x + 7 \) and shade below it.

System of Inequalities

A system of inequalities consists of multiple inequalities that are considered simultaneously. To find a solution set for a system, we need to look for the area where the shaded regions of all the individual inequalities overlap. In essence, it's like solving puzzles where each inequality gives a piece of the overall picture.

The overlapping region, if it exists, will be on every line or on one side of every line in the system, depending on whether the inequalities are 'less than' or 'greater than'. Working through these systems develops a deep understanding of how multiple constraints can influence a solution.

Algebraic Solutions

While graphing provides a visual representation, algebraic solutions to systems of inequalities require manipulating equations to find where they intersect, or if they intersect at all. This often involves expressing variables from one inequality in terms of the other and then solving for the remaining variable.

Alternatively, we can use methods like the elimination or substitution method, similar to solving systems of equations. However, unlike equations that yield specific coordinate points, solutions to inequalities are typically ranges or regions. For the provided exercise, we're focused on visually graphing, which provides immediate insights into the nature of these ranges.

Boundary Lines

The boundary lines in a system of inequalities are the dividing lines that separate the different areas of the graph. They represent the 'equals' part in 'less than or equal to' (\(\leq\)) and 'greater than or equal to' (\(\geq\)) inequalities.

In our task, for instance, the first inequality \(y \< -3x +3 \) has the boundary line \(y = -3x +3\). Where the boundary line is dashed (\(y < -3x +3\)), the points along the line are not included in the solution set. Where it's solid (\(y \= -3x +3\)), they are included. The boundary lines guide us in determining the correct area to shade, illustrating the potential solution set for the system.