/*! 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 34 Solve the inequality. Express th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 34

Solve the inequality. Express the answer using interval notation. $$ |x-a|

Short Answer

Expert verified
\( (a-d, a+d) \)

Step by step solution

01

Understanding Absolute Value Inequality

The inequality \(|x-a| < d\) represents a range of values for \(x\) that are within a distance \(d\) from \(a\). This implies that the expression \(|x-a|\) can be rewritten as the compound inequality: \[ -d < x-a < d \]
02

Isolating x

To solve for \(x\), we need to isolate \(x\) in the compound inequality. We'll do this by adding \(a\) to all parts of the inequality.\[ -d + a < x < d + a \]
03

Writing in Interval Notation

The solution \(-d + a < x < d + a\) can be expressed in interval notation as:\((a-d, a+d)\). This interval includes all values of \(x\) greater than \(a-d\) and less than \(a+d\).

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 Inequality
An absolute value inequality involves an expression of the form \(|x-a| < d\). Here, the absolute value expression \(|x-a|\) measures the distance of \(x\) from \(a\). \(d\) is the maximum allowable distance, and the inequality states that \(x\) must be within this distance from \(a\).
  • For any number inside the absolute value, say \(b\), \(|b|\) represents how far \(b\) is from zero, regardless of direction.
  • In the inequality \(|x-a| < d\), \(x\) can take any value that makes the distance from \(a\) less than \(d\).
Understanding this concept helps us rewrite the absolute value inequality into a more manageable form, specifically a compound inequality, and solve it step by step.
Interval Notation
Interval notation is a shorthand method of expressing the set of solutions for an inequality. For the inequality \-d < x-a < d\, which is derived from the absolute value inequality, we rephrase it into the simpler interval notation form.
  • This form uses parentheses or brackets to indicate whether endpoints are included (closed intervals) or not (open intervals).
  • For the inequality \-d+a < x < d+a\, the solution in interval notation will be \( (a-d, a+d) \).
  • The parentheses indicate that the endpoints \(a-d\) and \(a+d\) are not included in the solution set.
Interval notation is efficient and visually intuitive, showing the solution on a number line, letting us see at a glance which numbers satisfy the given conditions.
Compound Inequality
A compound inequality is an equation with two inequalities joined together. In our context, the absolute value inequality \(|x-a| < d\) becomes the compound inequality \(-d < x-a < d\).
  • This implies that the value of \(x\) must satisfy both conditions of the inequality simultaneously.
  • To solve it, follow the steps within each part of the inequality while maintaining the relations linked by the inequality sign.
  • Adding or subtracting the same number from all three parts of the inequality is a common technique used to isolate the variable \(x\).
In this scenario, by isolating \(x\) through algebraic manipulation, we arrive at the solution as \-d+a < x < d+a\, and express it seamlessly in interval notation as \( (a-d, a+d) \). Understanding compound inequalities facilitates solving complex problems by simplifying them into smaller, manageable parts.

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

\(67-70=\) Determine the values of the variable for which the expression is defined as a real number. $$ \left(\frac{1}{x^{2}-5 x-14}\right)^{1 / 2} $$

Driving Cost It is estimated that the annual cost of driving a certain new car is given by the formula $$ C=0.35 m+2200 $$ where \(m\) represents the number of miles driven per year and \(C\) is the cost in dollars. Jane has purchased such a car, and decides to budget between \(\$ 6400\) and \(\$ 7100\) for next year's driving costs. What is the corresponding range of miles that she can drive her new car?

Solve the equation. $$ |x+4|=0.5 $$

Bonfire Temperature In the vicinity of a bonfire, the temperature \(T\) in \(^{\circ} \mathrm{C}\) at a distance of \(x\) meters from the center of the fire was given by $$ T=\frac{600,000}{x^{2}+300} $$ At what range of distances from the fire's center was the temperature less than \(500^{\circ} \mathrm{C} ?\)

Suppose that \(a, b, c,\) and \(d\) are positive numbers such that $$ \frac{a}{b}<\frac{c}{d} $$ show that $$ \frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Statistics

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.