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Chapter 2: Problem 11

Graph each inequality, and write it using interval notation. \(x<-3\)

Short Answer

Expert verified
(\(-finity, -3\))

Step by step solution

01

Understand the inequality

The inequality given is \(x < -3\). This means that \(x\) can take any value that is less than \(-3\).
02

Draw the number line

Draw a horizontal line and mark the point \(-3\) on it. This will be our reference point for the inequality.
03

Indicate the region for the inequality

Shade the portion of the number line to the left of \(-3\). Use an open circle at \(-3\) to indicate that \(-3\) is not included in the solution.
04

Write the inequality in interval notation

In interval notation, \(x < -3\) is written as \((-finity, -3)\). The parenthesis indicates that \(-3\) is not included in the interval.

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Key Concepts

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

Number Line
The number line is a visual representation of numbers placed in a straight line. The line extends infinitely in both positive and negative directions, symbolizing all potential real numbers.
  • The center of the number line is typically marked by 0.
  • Positive numbers are marked to the right of 0.
  • Negative numbers are marked to the left of 0.
In this exercise, we are specifically looking at the point -3. Position -3 correctly on the number line and use it as a reference point for our inequality.
Remember to use open and closed circles to specify whether endpoints are included or excluded in the solution set.
Interval Notation
Interval notation is a method of writing subsets of the real number line by specifying the start and end points of an interval. The notation can tell if endpoints are included or not:
  • Use parenthesis \( ( \) or \( ) \) to denote that an endpoint is excluded. For example, \( (-3, 5) \) means all the numbers between -3 and 5, excluding -3 and 5 themselves.
  • Use brackets \( [ \) or \( ] \) to denote that an endpoint is included. For example, \( [-3, 5] \) means all numbers between -3 and 5, including both -3 and 5.

In our exercise, the inequality \( x < -3 \) is represented as \( (-\infty, -3) \). The negative infinity symbol \( -\infty \) indicates that the interval extends indefinitely to the left, and the parenthesis around -3 indicates that -3 is not included in the interval.
Inequalities
Inequalities are mathematical expressions involving the symbols \( < \), \( > \), \( \leq \), or \( \geq \), used to compare two values or expressions.
They tell us about the relative sizes of expressions and which values solve the inequality:
  • \( x < -3 \) means x is any number less than -3.
  • \( x \leq -3 \) includes -3 and all numbers less than -3.
  • \( x > -3 \) means x is any number greater than -3.
  • \( x \geq -3 \) includes -3 and all numbers greater than it.
Always remember to represent them correctly on a number line using open or closed circles to indicate excluded or included endpoints, respectively.
This helps us in visualizing where the solutions to an inequality lie on the number line.

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