/*! 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 69 Solve the absolute value inequal... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 69

Solve the absolute value inequality and express the solution set in interval notation. $$|4-x| \leq 1$$

Short Answer

Expert verified
The solution set in interval notation is [3, 5].

Step by step solution

01

Understanding the Absolute Value Inequality

The given absolute value inequality is \(|4-x| \leq 1\). This inequality can be split into two separate inequalities: \(4-x\leq1\) and \(4-x\geq-1\). These two conditions must be true simultaneously for the absolute value inequality to hold.
02

Solve the First Inequality

Solve the inequality \(4-x\leq1\):\[\begin{align*} 4-x & \leq 1 \ -x & \leq 1-4 \ -x & \leq -3 \ x & \geq 3 \quad \text{(Multiply or divide by a negative number, flips the inequality sign)} \end{align*}\]
03

Solve the Second Inequality

Solve the inequality \(4-x\geq-1\):\[\begin{align*} 4-x & \geq -1 \ -x & \geq -1-4 \ -x & \geq -5 \ x & \leq 5 \quad \text{(Multiply or divide by a negative number, flips the inequality sign)} \end{align*}\]
04

Combine the Solution

Combining the two inequalities from steps 2 and 3, we have \(3 \leq x \leq 5\). This can be written in interval notation as \([3, 5]\).

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.

Interval Notation
Interval notation is a way to describe the set of solutions for inequalities. It provides a simplified way to express ranges of numbers. Instead of writing out all the elements in a solution set, we use brackets to denote the boundaries.

For the inequality solution \(3 \leq x \leq 5\), we place both numbers inside square brackets to indicate that both 3 and 5 are included in the solution. This gives us the interval \([3, 5]\).

Here are some key points to remember:
  • Square brackets \([ ]\) mean the endpoint is included (also known as closed interval).
  • Parentheses \(( )\) mean the endpoint is not included (also known as open interval).
  • Example of an open interval: \((3, 5)\) where 3 and 5 are not included.
  • Mixed intervals, like \([3, 5)\), mean 3 is included while 5 is not.
Using interval notation makes it easier to work with sets of real numbers, especially when solving inequalities.
Solving Inequalities
Understanding how to solve inequalities is crucial when working with absolute value inequalities. The key difference between an equation and an inequality is that inequalities deal with a range of solutions rather than a single value.

When solving inequalities, it's important to follow certain rules:
  • When you add or subtract the same number on both sides, the inequality sign remains the same. For example, if \(x + 2 \leq 4\), then subtracting 2 from both sides gives \(x \leq 2\).
  • However, if you multiply or divide both sides of the inequality by a negative number, the inequality sign flips. For instance, \(-x \leq -3\) becomes \(x \geq 3\) when multiplying by -1.
  • When splitting an absolute value inequality into two cases, ensure both separate inequalities are solved independently before combining their solutions.
With these principles in mind, solving the two inequalities from the absolute value inequality \(|4-x| \leq 1\) provides the solution set \([3, 5]\). This means any number between 3 and 5, including the endpoints, is a solution.
Absolute Value Properties
The concept of absolute value relates to the distance of a number from zero on a number line. It鈥檚 always non-negative because distance cannot be negative.

The absolute value operation has special properties that become useful when solving inequalities:
  • For any number \(a\), \(|a| \geq 0\). This means the absolute value of any expression is always zero or positive.
  • An equation of the form \(|x| = a\) can be rewritten as two separate equations: \(x = a\) or \(x = -a\).
  • For inequalities, \(|x| \leq a\) is rewritten as \(-a \leq x \leq a\), and \(|x| \geq a\) becomes \(x \leq -a\) or \(x \geq a\).
In the context of the original problem \(|4-x| \leq 1\), recognizing that the inequality expresses a distance \(4-x\) within 1 unit of zero allows us to split it into two simpler inequalities: \(4-x \leq 1\) and \(4-x \geq -1\). This approach greatly simplifies solving the inequality.

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

Home Improvement. The cost of having your bathroom remodeled is the combination of material costs and labor costs. The materials (tile, grout, toilet, fixtures, etc.) cost is \(\$ 1,200\) and the labor cost is \(\$ 25\) per hour. Write an equation that models the total cost \(C\) of having your bathroom remodeled as a function of hours \(h\). How much will the job cost if the worker estimates 32 hours?

Determine whether the lines are parallel, perpendicular, or neither, and then graph both lines in the same viewing screen using a graphing utility to confirm your answer. $$\begin{aligned} &y_{1}=0,16 x+2.7\\\ &y_{2}=6.25 x-1.4 \end{aligned}$$

Refer to the following: Einstein's special theory of relativity states that time is relative: Time speeds up or slows down, depending on how fast one object is moving with respect to another. For example, a space probe traveling at a velocity \(v\) near the speed of light \(c\) will have "clocked" a time \(t\) hours, but for a stationary observer on Earth that corresponds to a time \(t_{0} .\) The formula governing this relativity is given by $$ t=t_{0} \sqrt{1-\frac{v^{2}}{c^{2}}} $$ If the time elapsed on a space probe mission is 5 years but the time elapsed on Earth during that mission is 30 years, how fast is the space probe traveling? Give your answer relative to the speed of light.

Temperature. The National Oceanic and Atmospheric Administration (NOAA) has an online conversion chart that relates degrees Fahrenheit, \(^{\circ} \mathrm{F}\), to degrees Celsius, \(^{\circ} \mathrm{C} .77^{\circ} \mathrm{F}\) is equivalent to \(25^{\circ} \mathrm{C},\) and \(68^{\circ} \mathrm{F}\) is equivalent to \(20^{\circ} \mathrm{C}\). Assuming the relationship is linear, write the equation relating degrees Celsius to degrees Fahrenheit. What temperature is the same in both degrees Celsius and degrees Fahrenheit?

Solve each formula for the specified variable. $$P=2 l+2 w \text { for } l$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.