Problem 80 - Precalculus

Question

In Exercises \(79-84,\) determine whether each statement is true or false. A dashed curve is used for strict inequalities.

Step-by-Step Solution

Answer

True.

Step 1: Understanding the Statement

The statement claims that a dashed curve is used to represent strict inequalities on a graph. In graphing inequalities, strict inequalities refer to those without equality, specifically `<` or `>`. These inequalities do not include the boundary line; hence, a visual separation is needed between what is included and what is not.

Step 2: Analyzing Graphing Conventions

In graphing conventions for inequalities: 1. A solid line or curve is used when the inequality includes equality (e.g., `\( \leq \)` or `\( \geq \)`), indicating that the points on the line or curve satisfy the inequality. 2. A dashed line or curve is used for strict inequalities (`\( < \)` or `\( > \)`) to show that the points on the line are not included in the solution.

Step 3: Evaluating the Truth of the Statement

Given the conventions described, the statement is evaluating whether a dashed curve indeed corresponds to strict inequalities, namely `\( < \)` or `\( > \)`. Since these inequalities represent solutions that do not include the boundary line itself, the use of a dashed line or curve is appropriate and aligns with standard graphing practices.

Step 4: Conclusion

Since a dashed curve is typically used to indicate strict inequalities where the boundary is not part of the solution set, the original statement is consistent with common graphical representations. Hence, the statement is true.

Key Concepts

Strict Inequalities

Strict inequalities refer to relationships where one quantity is less than or greater than another. They are represented by the symbols \(<\) or \(>\). In mathematics, strict inequalities indicate that the boundary itself is not part of the solution set. This means that if the inequality is \(x < 5\), then 5 is not included in the possible values for \(x\). Strict inequalities often require clear visual distinctions, especially when graphing. Unlike non-strict inequalities such as \(x \leq 5\), where 5 would be included, strict inequalities are used to show that none of the boundary line points belong to the set of solutions. This helps in visually interpreting whether or not points on the line are part of what defines the inequality. When solving problems involving strict inequalities, it is crucial to remember that the solution set will never have the boundary point itself; it always falls strictly either below or above, purely within the range defined by the inequality.

Dashed Curve

In the context of graphing inequalities, a dashed curve is an essential part of visual representation. Its primary role is to illustrate strict inequalities, specifically those described by \(<\) or \(>\) symbols. But why a dashed curve?

• It visually suggests separation. Unlike a solid line, a dashed curve clearly shows that points along this line are not part of the solution set.
• It helps distinguish strict inequalities from non-strict ones. By using a dashed line, you immediately identify the inequality as either strictly less than or greater than, where boundary points are excluded.

When graphing a region that satisfies a strict inequality, the dashed curve acts as a powerful visual aid, providing clarity about what is and isn’t included in the set of solutions. This is especially useful in visual mathematics where distinctions must be clear.

Solid Line

A solid line in graphing is a contrasting visual tool to a dashed line, used when graphing non-strict inequalities. Non-strict inequalities include the boundary in their solution set and are represented using \( \leq \) or \( \geq \) symbols.

• The solid line indicates that the boundary values are part of the solution. If you graph \(y \leq 2x \- 3\), every point on the line \(y = 2x \- 3\) is included in the solution set.
• Solid lines visually communicate that both sides of the boundary are possibly part of the solution, suggesting an inclusive relationship.

This type of line is perfect for inequalities that consider boundary conditions, providing a straightforward way to denote inclusion in graphs and contributing to accurate interpretations of mathematical expressions.

Boundary Line

Graphically, the boundary line is a crucial component in the representation of inequalities. It acts as a separating factor in the plane, distinguishing regions where the inequality holds true from those where it doesn't.

• For strict inequalities, the boundary line is dashed, indicating exclusion of all points lying on it. This aligns with the principles of strict inequalities where boundaries aren't part of solutions.
• In non-strict inequalities, the boundary line is solid, indicating inclusion, meaning all points on the line satisfy the inequality too.

Understanding the function of a boundary line is essential to interpreting inequalities accurately. It not only points out the limit or threshold but determines the method of graphing itself—whether to use a dashed or a solid line depending on the nature of the inequality involved. This clear demarcation fosters better understanding and helps avoid common mistakes when solving graphing problems.