Problem 31 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \left\\{\begin{array}{l} y>2 x-3 \\ y<-x+6 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution set for the system of inequalities is the area that lies below the line \(y = -x + 6\) and above the line \(y = 2x - 3\).

Step 1 - Graph the first inequality

Plot the line \(y = 2x - 3\). This line is not included in the solution because the inequality is \(y > 2x - 3\), so draw it as a dashed line. The solution area should be above the line as \(y > 2x - 3\).

Step 2 - Graph the second inequality

Next, plot the line \(y = -x + 6\). Since the inequality is \(y < -x + 6\), also draw this one as a dashed line. The solution area should be below the line as \(y < -x + 6\).

Step 3 - Find the intersection

The intersection of the shaded regions from step 1 and step 2 gives the solution set for the system of inequalities. That is the area below the line \(y = -x + 6\) and above the line \(y = 2x - 3\).

Key Concepts

Inequality Solutions

Understanding inequality solutions is crucial for solving systems of inequalities. An inequality solution refers to the set of all possible values that satisfy the inequality. Unlike equations, which typically have a single solution or a set of discrete solutions, an inequality may have a range of continuous solutions.

For instance, in the given system, the individual inequalities are given as:

• \(y > 2x - 3\)
• \(y < -x + 6\)

To find the solution set for this system, we must identify where the solutions for both inequalities overlap. The overlapping region on a graph represents all the points \((x, y)\) that satisfy both inequalities simultaneously. In graphical terms, you're looking for the region that is above the line \(y = 2x - 3\) and below the line \(y = -x + 6\). This region, if one exists, is often a polygon-shaped area where the system's constraints are all met.

Linear Inequalities

A linear inequality looks similar to a linear equation but uses inequality symbols (\(>, <, \geq, \leq\)) instead of an equals sign. When graphing linear inequalities, we graph the associated line but treat the inequality portion in a unique way.

If the inequality is 'less than' (\(<\)) or 'greater than' (\(>\)), we draw a dashed line, indicating that the line itself is not included in the solution set. For 'less than or equal to' (\(\leq\)) or 'greater than or equal to' (\(\geq\)), we draw a solid line since points on the line satisfy the inequality.

Afterward, we select a test point, not on the line (usually the origin is a good choice if it's not on the line), to determine which side of the line is part of the solution set. If the test point satisfies the inequality, then the area containing the test point is shaded; if not, the other side is shaded.

Algebraic Graphing

The term algebraic graphing involves plotting algebraic equations or inequalities on a coordinate plane to visually represent their solutions. When working with systems of inequalities, algebraic graphing helps to find the solution set by showing the common shaded area where all inequalities are true.

To graph a linear inequality:

• Graph the corresponding equation of the boundary line. Use dashed lines for '<' or '>' inequalities, and solid for '\(\leq\)' or '\(\geq\)'.
• Select a test point that lies outside the boundary line, typically \((0, 0)\) unless it lies on the line itself.
• If the test point satisfies the inequality, shade the side of the line where the test point lies. Otherwise, shade the opposite side.

In our exercise, both inequalities are graphed to find their respective solution areas. The intersection of the solution areas for each inequality is the solution to the system. This method allows us to represent and solve complex relationships visually, making algebraic concepts more accessible and understandable.