/*! 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 113 Solve each equation or inequalit... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 113

Solve each equation or inequality. $$ |x-2|+3 \geq 2 $$

Short Answer

Expert verified
The solution is all real numbers, \((-\infty, \infty)\).

Step by step solution

01

- Isolate the Absolute Value Expression

First, isolate the absolute value expression by subtracting 3 from both sides of the inequality: \[|x - 2| + 3 - 3 \geq 2 - 3 \] which simplifies to \[ |x - 2| \geq -1 \]
02

- Consider Properties of Absolute Value

Recognize that the absolute value of any expression is always greater than or equal to 0. Therefore, \[ |x - 2| \geq -1 \] is always true for all real numbers.
03

- Determine the Solution Set

Since the inequality is always true for all real numbers, the solution set is all real numbers, which can be represented as \( (-\infty, \infty) \).

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.

absolute value
The concept of absolute value is essential in various branches of mathematics. The absolute value of a number represents its distance from zero on the number line, regardless of direction. To denote the absolute value of a variable, we use vertical bars. For example, the absolute value of \( x \) is written as \( |x| \).

Absolute values are always non-negative, as distances can never be negative. This means \( |x| \geq 0 \) for any real number \( x \).

When working with expressions within absolute value symbols, remember:
  • \( |x| = x \) if \( x \) is positive or zero.
  • \( |x| = -x \) if \( x \) is negative.
inequality solutions
Solving inequalities involving absolute values follows specific steps to determine the range of possible values for the variable. Initially, isolate the absolute value expression on one side of the inequality.

For example, given \[ |x - 2| + 3 \geq 2 \] we first subtract 3 from both sides of the inequality:
\[ |x - 2| + 3 - 3 \geq 2 - 3 \]
which simplifies to:
\[ |x - 2| \geq -1 \]

This highlights another key point: the absolute value of an expression is always non-negative. Hence, \(|x - 2| \geq -1 \) is always true for all real numbers.
The solution set includes all real numbers, represented as \( (-\infty, \infty) \).
real numbers
Real numbers encompass a wide range of numbers including:
  • Positive and negative integers (e.g., -3, 4)
  • Rational numbers (e.g., fractions like \( \frac{1}{2} \))
  • Irrational numbers (e.g., \( \pi \), \( \sqrt{2} \))


When solving inequalities like \( |x - 2| \geq -1 \), it鈥檚 important to understand that absolute values tie into this concept, as they can take on any real number value.

In our example, since the absolute value expression is always non-negative, we conclude that the inequality holds for all real numbers.
algebraic expressions
An algebraic expression can include variables, numbers, and arithmetic operations. Understanding how to manipulate these expressions is vital for solving inequalities.

For the inequality \( |x - 2| + 3 \geq 2 \), combining like terms or isolating the absolute value expression requires algebraic manipulation.

Here鈥檚 how we approached it:
  • Subtracted 3 from both sides: \( |x - 2| + 3 - 3 \geq 2 - 3 \).
  • This simplified to: \( |x - 2| \geq -1 \).
This manipulation helps to bring the absolute value into focus,
ensuring we apply the necessary mathematical rules to solve the inequality for the variable x.

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

Solve each equation or inequality. $$ |x| \geq-15 $$

The recommended daily intake (RDI) of calcium for females aged \(19-50\) is \(1000 \mathrm{mg}\). Write this statement as an absolute value inequality, with \(x\) representing the RDI, to express the RDI plus or minus \(100 \mathrm{mg}\). Solve the inequality. (Data from National Academy of Sciences - Institute of Medicine.)

Solve each equation or inequality. $$ |x+5|-6 \leq-1 $$

The 10 tallest buildings in Houston, Texas, are listed along with their heights. $$ \begin{array}{|l|c|} \hline \quad {\text { Building }} & \text { Height (in feet) } \\ \hline \text { JPMorgan Chase Tower } & 1002 \\ \text { Wells Fargo Plaza } & 992 \\ \text { Williams Tower } & 901 \\ \text { Bank of America Center } & 780 \\ \text { Texaco Heritage Plaza } & 762 \\ \text { 609 Main at Texas } & 757 \\ \text { Enterprise Plaza } & 756 \\ \text { Centerpoint Energy Plaza } & 741 \\ \text { 1600 Smith St. } & 732 \\ \text { Fulbright Tower } & 725 \\ \hline \end{array} $$ Use this information To find the average of a group of numbers, we add the numbers and then divide by the number of numbers. Use a calculator to find the average of the heights.

Solve each equation or inequality. $$ |10 x+7|+3<1 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Applied Mathematics

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.