/*! 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 83 Solve each inequality or compoun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 83

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ y-\frac{1}{7} \leq \frac{2}{3} $$

Short Answer

Expert verified
The solution is \((-\infty, \frac{17}{21}]\).

Step by step solution

01

Isolate the variable

To solve the inequality \( y - \frac{1}{7} \leq \frac{2}{3} \), we need to isolate \( y \). Start by adding \( \frac{1}{7} \) to both sides of the inequality to undo the subtraction. This gives us: \[ y \leq \frac{2}{3} + \frac{1}{7} \].
02

Find common denominator

To combine the fractions \( \frac{2}{3} \) and \( \frac{1}{7} \), we need a common denominator. The least common multiple of 3 and 7 is 21. Convert the fractions to have this denominator: \( \frac{2}{3} = \frac{14}{21} \) and \( \frac{1}{7} = \frac{3}{21} \).
03

Add the fractions

Now add the fractions \( \frac{14}{21} + \frac{3}{21} \). This results in \( \frac{17}{21} \). Translated to the inequality, we have: \[ y \leq \frac{17}{21} \].
04

Express solution in interval notation

The solution to the inequality \( y \leq \frac{17}{21} \) can be expressed in interval notation. Since \( y \) can take any value less than or equal to \( \frac{17}{21} \), the interval is \((-\infty, \frac{17}{21}]\).
05

Graph the solution

On a number line, plot the solution. Draw a solid dot at \( \frac{17}{21} \) to indicate that this value is included in the solution. Then, shade the line to the left of this point to indicate all values less than \( \frac{17}{21} \) are part of the solution.

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
When solving inequalities, interval notation is a concise way to express all possible solutions. It uses brackets and parentheses to indicate which numbers are part of the solution. In our exercise, the inequality solution was \( y \leq \frac{17}{21} \).
To express this in interval notation, we needed to show all the values \( y \) can take, which are any numbers less than or equal to \( \frac{17}{21} \).
  • A bracket \([a, b]\) indicates that both \(a\) and \(b\) are included in the solution.
  • A parenthesis \((a, b)\) means \(a\) and \(b\) are not included.
For example, our solution \((-\infty, \frac{17}{21}]\) uses a parenthesis before negative infinity \(-\infty\), because infinity is not a specific number and cannot be included. However, the bracket at \( \frac{17}{21} \) shows this number is part of the solution set.
Common Denominator
Finding a common denominator is crucial when you need to add or compare fractions. In our inequality, before we could find the solution, we had to handle two fractions: \( \frac{2}{3} \) and \( \frac{1}{7} \).
These fractions need a common denominator to be combined. Here's how you do it:
  • Identify the denominators you are working with 鈥� here, they are 3 and 7.
  • Determine the least common multiple (LCM) of the denominators. For 3 and 7, this is 21.
  • Convert each fraction to an equivalent fraction with the common denominator.\[ \frac{2}{3} \to \frac{14}{21} \text{ and } \frac{1}{7} \to \frac{3}{21} \]
Once they are over the same denominator, it becomes simple to add them up. This step made it possible to find our inequality solution \( y \leq \frac{17}{21} \).
Number Line Graphing
Graphing inequalities visually shows the solution set on a number line. It's especially useful in seeing the range of values included in the solution.
In the exercise, after solving the inequality and writing \( y \leq \frac{17}{21} \), we needed to graph it on a number line.
  • Locate \( \frac{17}{21} \) on the number line.
  • Draw a solid dot at \( \frac{17}{21} \) to signify that this endpoint is included in the solution.
  • Shade the number line to the left of this point, indicating all values less than \( \frac{17}{21} \) are solutions.
The solid dot at the end is crucial. It distinguishes values that are part of the solution from those that are not, showing equality \( \leq \) in the inequality. This visual tool helps students to grasp the full scope of the solution set and its boundaries.

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

Room Temperatures. To hold the temperature of a room between \(19^{\circ}\) and \(22^{\circ}\) Celsius, what Fahrenheit temperatures must be maintained? Hint: Use the formula \(C=\frac{5}{9}(F-32)\)

Number Puzzles. What numbers satisfy the condition: Four more than three times the number is at most \(10 ?\)

Dry mixture problems. How many scoops of natural seedless raisins costing 3.45 dollar per scoop must be mixed with 20 scoops of golden seedless raisins costing 2.55 dollar per scoop to obtain a mixture costing 3 dollar per scoop?

Dry mixture problems. When a child emptied his coin bank, he had a collection of pennies, nickels, and dimes. There were 20 more pennies than dimes and the number of nickels was triple the number of dimes. If the coins had a value of 5.40 dollar, how many of each type coin were in the bank?

Complete each table. $$ \begin{array}{|c|c|} \hline x & {\frac{x}{3}+2} \\ \hline-6 & {} \\ \hline 0 & {} \\ \hline 12 & {} \\ \hline \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

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.