Problem 26 - Precalculus

Question

In Exercises \(21-50,\) graph each system of inequalities or indicate that the system has no solution. $$\begin{aligned} &y > 2 x\\\ &y < 2 x \end{aligned}$$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Understand the System of Inequalities

The system consists of two linear inequalities. The first inequality is \( y > 2x \), and the second inequality is \( y < 2x \). We need to graphically represent the solution for both inequalities.

Step 2: Graph the Boundary Lines

For both inequalities, the boundary line is \( y = 2x \). Draw this line on the graph. It is important to remember that since both inequalities are strict (\(>\) and \(<\)), the line itself won't be part of the solution, so it should be drawn as a dotted line.

Step 3: Shade the Region for Each Inequality

Shade the region above the line \( y = 2x \) to represent all points where \( y > 2x \). Shade the region below the line \( y = 2x \) to show all points where \( y < 2x \). Each inequality, when graphed separately, will have a distinct shaded region.

Step 4: Identify the Solution Region

The solution to the system of inequalities is the area where the shaded regions for each inequality overlap. Since one inequality requires \( y \) to be greater than \( 2x \) and the other requires it to be less than \( 2x \), there is no overlap. Thus, the system has no solution.

Key Concepts

graphing inequalities

Graphing inequalities can be understood by first graphing the associated linear equation, and then shading the relevant region. For instance, if we have the inequality \( y > 2x \), this means that instead of considering just the line \( y = 2x \), we consider everything above it. Here's how it works:

• Draw the boundary line: Start by graphing the line \( y = 2x \). Because the inequality uses "greater than" (\(>\)), the line itself is not included. So, use a dotted line instead of a solid one.
• Choose a test point: Select a simple point that is not on the line, such as the origin (0,0). Substitute this point into the inequality to see if it makes the inequality true. In this case, \( 0 > 2 \times 0 \) is not true, so the origin is not in the solution region. Hence, shade the region opposite the origin.
• Shade the correct region: If the test confirms, shade the region where the inequality holds true. Remembering which side to shade can be tricky, but testing a point helps decide.

It's crucial to keep in mind, especially for systems of inequalities, which shaded areas are part of the solution. Dotted lines help distinguish boundaries not included in the solution.

linear inequalities

Linear inequalities are similar to linear equations but include inequality symbols like \( >, <, \geq,\) or \(\leq\) instead of an equals sign. These inequalities graphically represent regions rather than just lines. Let’s explore more:

• Equation to Inequality: Begin with an equation, e.g., \( y = 2x \), then convert it into an inequality by switching the \( = \) to \( <, >, \leq,\) or \(\geq \). This changes the entire nature of the solution from a line to an area.
• Inclusion of the Line: Remember, if the inequality includes "equal to" parts such as \( \leq \) or \(\geq \), the boundary line becomes solid as the line itself is part of the solution.
• Strict vs. Non-strict Inequalities: For strict inequalities (< or >), the boundary is not part of the solution, indicated by a dotted line. For non-strict inequalities (\(\leq \) or \(\geq \)), use a solid line. This distinction is critical in determining whether points on the line satisfy the inequality.

Understanding these details lets you visualize the entire set of possible solutions that satisfy the given inequality, rather than a strict set of points.

solution of inequalities

In systems of inequalities, the solution is the collection of points that satisfy all inequalities in the system simultaneously. Here's how to approach finding these solutions:

• Graph each inequality: Plot each one on the same graph, following steps outlined for graphing single inequalities. Each will have its respective shaded region.
• Find common areas: The solution for a system occurs where the shaded regions from individual inequalities overlap. This visual intersection represents points satisfying each inequality condition.
• No overlap means no solution: If there's no common region, like in our problem, the system has no solution. This occurs when conditions of the inequalities are contradictory.

Systems can range from having unique regions, common sections, or no shared space at all. By graphically analyzing these interactions, the underlying relationships between the inequalities are uncovered.