/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Solve each of the inequalities a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 22

Solve each of the inequalities and graph the solution set on a number line. \(4 x \geq-4\)

Short Answer

Expert verified
The solution is \(x \geq -1\), graphed as a closed circle on \(-1\) and shading to the right.

Step by step solution

01

Isolate the Variable

To solve the inequality, start by isolating the variable \(x\) on one side of the inequality. You can do this by dividing both sides of the inequality \(4x \geq -4\) by 4. This simplifies to \(x \geq -1\).
02

Verify the Solution Set

Check that the operation performed is correct and valid. Division by the positive number 4 does not change the direction of the inequality, so the solution is \(x \geq -1\).
03

Graph the Solution Set

Draw a number line, and indicate the point \(-1\) with a closed circle to show that this point is included in the solution set. Shade the number line to the right of \(-1\) to represent all numbers greater than or equal to \(-1\).

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.

Solving Inequalities
Inequalities are similar to equations, but instead of expressing equality, they show that one side is greater than or less than the other.
To solve an inequality like \(4x \geq -4\), the goal is to isolate the variable, which in this case is \(x\). This involves performing the same mathematical operations on both sides of the inequality.
Keep in mind the following rules when solving inequalities:
  • If you add or subtract the same number from both sides, the direction of the inequality stays the same.
  • If you multiply or divide both sides by a positive number, the inequality direction remains unchanged.
  • If you multiply or divide both sides by a negative number, you need to reverse the inequality sign.
In our example, we divide both sides by 4, a positive number, maintaining the inequality's direction. This simplifies \(4x \geq -4\) to \(x \geq -1\), indicating that \(x\) can be any number greater than or equal to \(-1\).
Graphing Inequalities
Once you have solved an inequality, it is helpful to graph the solution. This visual representation can make understanding the solution set easier.
For the inequality \(x \geq -1\), we represent the solution on a number line. Here鈥檚 how to graph inequalities:
  • Identify the key number (e.g., -1 in \(x \geq -1\)).
  • Use a closed circle to indicate that this number is included in the solution set (greater than or equal to).
  • If the inequality were strictly greater than (\(>\)) or less than (\(<\)), an open circle would be used instead, symbolizing that the key number isn鈥檛 included.
Placing a closed circle at -1 shows that -1 is part of the solution set, and shading to the right represents all numbers greater than or equal to -1.
Number Line Representation
A number line is a simple visual tool used to show the solution set of an inequality, providing a clear picture of which numbers satisfy the inequality.
For the inequality \(x \geq -1\), place a number line below the inequality.
Here are the steps for number line representation:
  • Draw a horizontal line with evenly spaced marks labeled with integers (e.g., -3, -2, -1, 0, 1, 2, 3).
  • Find and mark the critical value from the inequality on the number line. In this case, mark \(-1\).
  • Use a closed circle on -1 to show \(-1\) is part of the solution.
  • Shade the section of the line that represents the solution set. For \(x \geq -1\), shade to the right of -1, indicating that any number -1 or greater satisfies the inequality.
This representation helps visually confirm the solution and verify that the correct numbers have been included.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -1 \leq \frac{x+2}{4} \leq 1 $$

For Problems \(1-16\), solve each equation. $$ |x+2|-6=-2 $$

Solve each inequality. $$ \left|\frac{3 x-1}{4}\right|>3 $$

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ x+3>-2 \text { and } x+3<2 $$

For Problems \(17-30\), solve each inequality and graph the solution. $$ |x+1|>3 $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.