/*! 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 37 Solve. Write the solution set us... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 37

Solve. Write the solution set using interval notation. See Examples 1 through 7. $$ -x<-4 $$

Short Answer

Expert verified
The solution set is \((4, \, \infty)\).

Step by step solution

01

Understand the Inequality

The given inequality is \(-x < -4\). Our goal is to solve for \(x\). Notice that the inequality involves a negative coefficient on the variable \(x\).
02

Isolate x

To solve for \(x\), first eliminate the negative sign by multiplying every term of the inequality by -1. Remember that multiplying or dividing the inequality by a negative number will flip the inequality sign. Thus, \(-x < -4\) becomes \(x > 4\).
03

Write the Solution in Interval Notation

The inequality \(x > 4\) means \(x\) can be any number greater than 4. In interval notation, this is written as \((4, \, \infty)\). This notation indicates that 4 is not included in the solution set, but any number greater than 4 is included.

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 of expressing a range of values, particularly when solving inequalities. It's a concise method used in mathematics to specify the set of numbers that make up the solution. In interval notation:
  • Parentheses \( () \) indicate that a number is not included in the interval. For instance, \( (4, \, \infty) \) means "greater than 4." Here, 4 is not part of the solution set, but all numbers greater than 4 are.
  • Brackets \( [ ] \) are used when a number is included. For example, \( [4, \, \infty) \) would include 4 as part of the solution set, which is not applicable here since we have \( x > 4 \).
  • The symbol \( \infty \) (infinity) represents an unbounded limit, meaning there is no upper bound in the interval \( (4, \, \infty) \).
Understanding how to correctly use these notations is key in effectively communicating the range of solutions for inequalities.
Negative Coefficients
When you come across inequalities involving negative coefficients, like \(-x\), it鈥檚 important to carefully handle these terms when solving the inequality.
  • The inequality \(-x < -4\)\, involves a negative coefficient for \(x\), which means \(x\) is being subtracted rather than added, or can be seen as multiplied by -1.
  • To isolate \(x\), you can eliminate the negative coefficient by multiplying every part of the inequality by -1.
This step is crucial to simplify the equation into a more manageable form, as it converts \(-x\) into \(x\), giving you a positive term to work with. Handling negative coefficients properly ensures accurate solutions for \(x\). In this case, multiplying by -1 changes \(-x < -4\) to \(x > 4\).
Inequality Sign Reversal
A pivotal rule when manipulating inequalities is the reversal of the inequality sign. This occurs when you multiply or divide each side of the inequality by a negative number.
  • The inequality \(-x < -4\)\, serves as an example: multiplying both sides by -1 not only cancels the negative coefficient but also reverses the inequality sign, changing it to \(x > 4\).
  • The reasoning behind this is logical. Consider that if \(-x\) is less than \(-4\), flipping the inequality with multiplication by a negative aligns the inequality direction appropriately with numerical value order.
Always remember, the inequality direction must be adjusted each time a negative factor comes into play, ensuring your solution remains valid.

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 inequality. Graph the solution set and write it in interval notation. $$ |1+0.3 x| \geq 0.1 $$

The calorie count of a serving of food can be computed based on its composition of carbohydrate, fat, and protein. The calorie count \(C\) for a serving of food can be computed using the formula \(C=4 h+9 f+4 p,\) where \(h\) is the number of grams of carbohydrate contained in the serving, \(f\) is the number of grams of fat contained in the serving, and p is the number of grams of protein contained in the serving. Solve this formula for \(f,\) the number of grams of fat contained in a serving of food.

Consider the equation \(3 x-4 y=12 .\) For each value of \(x\) or \(y\) given, find the corresponding value of the other variable that makes the statement true. If \(x=2,\) find \(y\)

Solve each equation or inequality for \(x\). $$ |9+4 x| \geq 0 $$

Solve: \(2 x-3=5\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Pure 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.