/*! 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 28 Solve the inequality. Then graph... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 28

Solve the inequality. Then graph the solution set on the real number line. \(\frac{1}{x}<4\)

Short Answer

Expert verified
The solution to the inequality \(\frac{1}{x}<4\) is \(x > \frac{1}{4}\). On a number line, this is represented by a parenthesis on \(\frac{1}{4}\) and an arrow pointing to the right towards \(\infty\).

Step by step solution

01

Clear the fraction

In order to remove the fraction from the inequality, multiply both sides by x, keeping in mind that the direction of inequality changes if x is negative. For x > 0, we get \(x * (\frac{1}{x}) < 4x\), simplifying to \(1 < 4x\). However, if x < 0, the inequality is reversed, giving \(1 > 4x\).
02

Solve the inequality

Dividing both sides by 4 in each inequality yields \(x > \frac{1}{4}\) when x > 0 and \(x < \frac{1}{4}\) when x < 0. The inequality \(x < \frac{1}{4}\) for \(x < 0\) doesn't contribute any new solutions, so it can be ignored. The final solution is \(x > \frac{1}{4}\).
03

Graph the solution

the solution \(x > \frac{1}{4}\) is represented on a number line by a parenthesis on \(\frac{1}{4}\), signifying that \(\frac{1}{4}\) itself is not included in the solutions, and an arrow pointing towards the right to \(\infty\), indicating all numbers greater than \(\frac{1}{4}\) are solutions.

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.

Real Number Line
The real number line is a visual representation of real numbers arranged in increasing order from left to right. Think of it as an infinite line extending in both directions, where each point corresponds to a unique real number.
In this exercise, we use the real number line to demonstrate the set of solutions for the inequality we have solved. The real number line helps us visualize these solutions effectively.Here are some key components when working with the real number line:
  • Points: Specific numbers or solutions are shown as points on the line.
  • Intervals: A range of numbers that satisfy an inequality is depicted as a portion of the line.
  • Direction: Arrows signify continuity of numbers in a positive, negative, or both directions.
In our solution, we denote the interval starting just after \( rac{1}{4}\), extending towards positive infinity, showing all values of \( x\) that meet our condition.
Inequality Solving
Solving inequalities involves finding the set of numbers that make the inequality true. This can be similar to solving equations, but with a few extra considerations.When working with inequalities, remember:
  • Direction: The inequality sign ("><", "鈮�", ">=", ">") indicates the relationship between the two sides.
  • Reverse Sign: Multiplying or dividing by a negative value reverses the inequality sign.
  • Simplify: Just like equations, simplify both sides to isolate the variable.
In our example, \(\frac{1}{x}<4\), we made sure to treat cases where \(x\) could change the inequality's direction, particularly when \(x < 0\). Always check these conditions to ensure a correct solution.
Fractional Inequalities
Fractional inequalities are inequalities that contain fractions. Solving them often requires special techniques to handle the fractions properly.To solve \(\frac{1}{x}<4\), we had to clear the fraction. We multiplied through by \(x\), remembering that:
  • Same Sign: Multiplying by a positive maintains the inequality direction.
  • Reverse Sign: Multiplying by a negative reverses it, especially relevant if \(x\) may be negative.
After clearing fractions, solve the resulting inequalities individually. In this case, note how carefully analyzing sign changes allowed accurate problem solving.
Graphing Solutions
Graphing the solution of inequality provides a visual representation to quickly understand which numbers satisfy the condition.When graphing \(x > \frac{1}{4}\):
  • Open Circle or Parenthesis: Use these to indicate that the value \(\frac{1}{4}\) itself is not included as a solution.
  • Arrow: Draw an arrow extending to the right to show all numbers greater than \(\frac{1}{4}\) are included.
By placing the parenthesis at \(\frac{1}{4}\) and drawing the arrow towards the right, you clearly depict the solution set. The real number line then reflects this understanding visually, making it easier to identify and verify solutions.

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 the inequality and write the solution set in interval notation. \((x-1)^{2}(x+2)^{3} \geq 0\)

Solve the inequality. Then graph the solution set on the real number line. \(x^{3}-4 x \geq 0\)

For \(\$ 1500\) to grow to more than \(\$ 1890\) in 3 years, what must the simple interest rate be?

The average price \(B\) (in dollars) of brand name prescription drugs from 1998 to 2005 can be modeled by \(B=6.928 t-3.45, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to 1998 . Use the model to find the year in which the price of the average brand name drug prescription exceeded \(\$ 75\).

Find the domain of the expression. \(\sqrt{81-4 x^{2}}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.