Problem 38 Graph the solution set of each i... [FREE SOLUTION]

91影视

Precalculus : Building Concepts and Connections

Revathi Narasimhan

$Math Studyset 91影视 Explanations$ Math

0 Edition

Chapter 1: Problem 38

Graph the solution set of each inequality on the real number line. $$|x| \leq 3$$

Short Answer

Expert verified

The solution set of the inequality $|x| \leq 3$ is the interval [-3,3] on the real number line.

Step by step solution

Translate the Absolute Value Inequality to a Standard Inequality

Since $|x|$ represents the distance of $x$ from 0, the inequality $|x| \leq 3$ represents all $x$ such that the distance between $x$ and 0 is less than or equal to 3. This inequality translates into the standard inequality $-3 \leq x \leq 3$. This means that $x$ can be any number from -3 to 3 inclusive.

Graph the Solution on the Number Line

Draw a number line and label it with the numbers from -4 to 4. Mark and darken the line segment from -3 to 3 inclusive to represent the solution set of the inequality. Make sure to use closed circles at -3 and 3 to indicate that these numbers are included in the solution set.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Real Number Line

The real number line is a visual representation of all possible real numbers, extending infinitely in both directions. Imagine it as a straight pathway that stretches left and right without end. Each point on this line corresponds to one real number.

When dealing with inequalities or absolute values, the real number line serves to show us exactly where numbers sit in relation to each other.

**Points:** Each number corresponds to a point, exact and precise on the line.
**Continuity:** No gaps exist between points; every real number has a place.
**Infinite Extent:** Extends infinitely, covering every possible real number both negatively and positively.

For the inequality \, $|x| \leq 3$, it helps to think of the real number line as a map. Here, you can easily see and interpret where the numbers from -3 to 3 are located, including all points in between.

Graphing Inequalities

Graphing inequalities on a number line transforms an abstract concept into something you can see and understand easily. It shows which numbers and sections of the number line satisfy the inequality's conditions.

For \, $|x| \leq 3$, our job is to visualize the numbers that fit this condition. This means figuring out which parts of the number line include solutions:

**Draw the Line:** Start with a horizontal line, marking numbers symmetrically around zero.
**Identify Boundaries:** We realize the inequality translates to two points, -3 and 3. These are our boundaries.
**Shade the Solution Area:** For this inequality, we'll highlight the entire segment from -3 to 3. This shows every possible solution. Closed circles at -3 and 3 indicate these endpoints are included in our solution (as allowed by the inequality being "less than or equal to").

This visual makes it simple to see how the inequality operates, showcasing the range of values that truly work.

Distance on the Number Line

Distance on a number line refers to how far apart two points are from each other. It can be thought of as the absolute amount of space between them, regardless of direction. In absolute value terms, it represents how far a number is from zero.

For \, $|x| \leq 3$, the concept of distance is critical:

**Zero as Reference Point:** The number 0 is our central point here. It's where distances start, and in our inequality, we are measuring how far numbers are from zero.
**Symmetrical Nature:** Any number at distance 3 from 0 will satisfy the inequality鈥攖his includes both negative and positive directions, hence \, $ -3 \leq x \leq 3 $.
**Understanding "Less Than or Equal To":** This inequality not only looks at those points strictly 3 units away but also all numbers within this range. This shows the nature of absolute value inequalities鈥攊t's about including everything at this proximity or closer to the center point, zero.

Thus, grasping distance on a number line allows us to convert and solve absolute value inequalities effectively.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Graph the solution set of each inequality on the real number line. $$|x| \leq 3$$

Short Answer

Step by step solution

Translate the Absolute Value Inequality to a Standard Inequality

Graph the Solution on the Number Line

Key Concepts

Real Number Line

Graphing Inequalities

Distance on the Number Line

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Decision Maths

Statistics

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.