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Chapter 1: Problem 40

Sketch the given set on a number line. $$\\{x | x>-3\\}$$

Short Answer

Expert verified
Shade the number line to the right of an open circle at -3.

Step by step solution

01

Identify the Inequality

The given set is defined by the inequality \(x > -3\). This means we are looking for all numbers greater than -3.
02

Locate -3 on the Number Line

Draw a number line and locate the point -3. This will be our reference point.
03

Determine the Boundary Condition

Since the inequality is strictly greater than (-3), we do not include -3 in the set. Represent this by drawing an open circle around -3.
04

Shade the Appropriate Region

Shade the region to the right of -3, indicating all numbers greater than -3.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inequalities
Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols like '<', '>', '鈮�', and '鈮�' to compare the values. For example, the inequality \( x > -3 \) means that x can be any number greater than -3. Unlike equations, inequalities do not give a single solution but rather a range of possible solutions.

Understanding inequalities is essential in solving problems in algebra, calculus, and real-world situations where conditions and limits need to be defined.
number line
A number line is a visual representation of numbers on a straight horizontal line. Each point on the line corresponds to a number. Positive numbers are plotted to the right of zero, while negative numbers are plotted to the left.

When plotting inequalities on a number line, it's important to locate the boundary point first. For example, in the inequality \( x > -3 \), -3 is our boundary. We use this boundary to visualize the set of possible solutions.

An open circle is used to indicate that the boundary point itself is not included in the solution set (e.g., for '>' or '<'). A closed circle means the boundary is included (e.g., for '鈮�' or '鈮�').
graphing inequalities
Graphing inequalities involves shading the appropriate region on a number line to represent all possible solutions. Here's a step-by-step guide to graph \( x > -3 \):
  • 1. Draw a straight horizontal line representing a number line and mark the point -3.
  • 2. Create an open circle around -3 to show that it's not included in the solution set.
  • 3. Shade the area to the right of -3 to indicate that the solutions are all numbers greater than -3.

The shaded region helps to easily identify the set of values that satisfy the inequality. This visual aid makes it simpler to understand the solution range and communicate it effectively.

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