Problem 39 - 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. $$\left\\{\begin{array}{l}x+2 y<6 \\\y>2 x-2 \\\y \geq 2\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is the overlapping shaded region from all three inequalities. If there is no such region, then the system has no solutions.

Step 1: Graph Inequality 1

The first inequality is \(x+2y<6\). To graph this inequality, first rearrange it to \(y < -0.5x + 3\) which is the form \(y = mx + c\) to enable us to sketch the line. Plot a line with slope -0.5 and y-intercept 3. It will be a dashed line since y-values are strictly less than -0.5x + 3. Shade below this line as for \(y < -0.5x + 3\), y values are smaller than the value of -0.5x + 3 at any given x.

Step 2: Graph Inequality 2

The second inequality is \(y>2x-2\). It is already in the form \(y=mx+c\), so sketch a line with the slope 2 and y-intercept -2. The line plotted will be a dashed line, as the y-values are strictly > 2x - 2. Shading will be above this line as for \(y > 2x - 2\), y values are bigger than the value of 2x - 2 at any given x.

Step 3: Graph Inequality 3

The third inequality is \(y \geq 2\). This is a horizontal line at y=2. As y-values are equal to or greater than 2, it is a solid line and shading occurs above the line.

Step 4: Common Solution Area

The solution to the system of inequalities is the overlapping region(shaded portion) from all three inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves representing regions on a coordinate plane that satisfy an inequality condition. For each inequality, you usually start by graphing its corresponding linear equation. This means finding the line defined by the equation, as if the inequality sign was an equal sign.

At first, we rearrange the inequality into the slope-intercept form, which is generally given as \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. This form makes it easy to draw the line on a graph.

• For strict inequalities (like \(>\) or \(<\)), draw a dashed line to indicate that points on the line are not included in the solution set.
• For non-strict inequalities (like \(\geq\) or \(\leq\)), draw a solid line to indicate that points on the line are included in the solution set.

This visual representation helps us identify solutions that satisfy each inequality condition.

Solution Set

The solution set of a system of inequalities is the region where all inequalities in the system overlap. In a coordinate plane, this means finding the areas where all shaded regions from each inequality intersect.

To graphically determine this region, you:

• Graph each inequality individually on the same coordinate plane.
• Use different shading for each inequality to showcase the areas that satisfy each condition.
• Identify the common shaded region where all inequalities overlap, serving as the solution set.

Remember that if no common region exists for all inequalities, the system has no solution.

Linear Inequalities

Linear inequalities are mathematical statements involving linear expressions separated by inequality symbols (\(<, \leq, >, \geq\)). These describe regions in the coordinate plane. Understanding linear inequalities requires knowledge of linear equations and how inequalities modify them.

• Each linear inequality tells you about a half-plane (area on one side of a line).
• The inequality sign indicates whether you want the region above or below the line.

For instance, in the inequality \(x + 2y < 6\), when transformed to \(y < -0.5x + 3\), the line divides the plane, and you shade below it to find the corresponding half-plane for the inequality.

Shading Regions

Shading regions refers to coloring the areas of the graph that satisfy the inequality. This visual aid is crucial for understanding which parts of the graph meet the conditions of each inequality.

• Start by graphing the line of each inequality, considering if it's dashed or solid.
• The region to shade is determined by testing a point or seeing if the inequality is satisfied above or below the line.

This technique highlights the possible solutions visually, making it easier to identify the solution set or the overlap for a system of inequalities.