Problem 57 Solve the inequality and sketch ... [FREE SOLUTION]

91影视

Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 57

Solve the inequality and sketch the solution set on a number line. $$-1

Short Answer

Expert verified

$ -4 < x < -1 $

Step by step solution

Break Down the Compound Inequality

Break the compound inequality into two separate inequalities: 1. $ -1 < x + 3 $ 2. $ x + 3 < 2 $

Solve the First Inequality

Solve $ -1 < x + 3 $ Subtract 3 from both sides: $ -1 - 3 < x $ or $ -4 < x $

Solve the Second Inequality

Solve $ x + 3 < 2 $ Subtract 3 from both sides: $ x < 2 - 3 $ or $ x < -1 $

Combine the Results

Combine the results of the two inequalities: $ -4 < x < -1 $

Sketch the Solution on a Number Line

Draw a number line and shade the region between -4 and -1, excluding the points -4 and -1. Use open circles at -4 and -1 to indicate that these points are not 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.

Compound Inequality

A compound inequality consists of two separate inequalities joined by 'and' or 'or'. In our exercise, the compound inequality is written as $-1 < x + 3 < 2$. This means the value of $x+3$ must satisfy both inequalities at the same time. It's like having two conditions that $x $ must meet. For this particular problem, you break it into two parts: $-1 < x + 3$ and $x + 3 < 2$.

Number Line

A number line helps visualize the solution of an inequality by showing which values of \ x \ make the inequality true. When sketching $-4 < x < -1 $ on a number line:

Draw a horizontal line and mark numbers evenly along it.
Mark the points -4 and -1 with open circles to indicate they aren鈥檛 included in the solution.
Shade the region between -4 and -1 to show all the solutions for \ x \ .

This shaded region represents all $ x $ values that satisfy both inequalities.

Inequality Solution

When solving inequalities, we follow similar steps as solving equations but with attention to the direction of inequality signs:

To isolate \ x \ , we add, subtract, multiply, or divide both sides by the same number.
Subtracting the same number keeps the inequality direction unchanged.
After solving individual parts, combine them to get the complete solution.

In our case, the solution $-4 < x < -1$ shows the range of \ x \ values that make both $-1 < x + 3$ and $x + 3 < 2$ true.
This combined result represents the final, full solution of the compound inequality.

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

Solve the inequality and sketch the solution set on a number line. $$-1

Short Answer

Step by step solution

Break Down the Compound Inequality

Solve the First Inequality

Solve the Second Inequality

Combine the Results

Sketch the Solution on a Number Line

Key Concepts

Compound Inequality

Number Line

Inequality Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Calculus

Theoretical and Mathematical Physics

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.