/*! 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 11 Solve each inequality. $$-2 y&... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 11

Solve each inequality. $$-2 y>-7$$

Short Answer

Expert verified
y < 3.5

Step by step solution

01

- Isolate the Variable

Start by isolating the variable on one side of the inequality. To do this, divide both sides of the inequality by -2.
02

- Reverse the Inequality Sign

Since you are dividing both sides by a negative number, reverse the direction of the inequality sign. This changes the inequality to: $$ y < \frac{7}{2}$$
03

- Simplify

Simplify the right-hand side to get the final answer: $$ y < 3.5 $$

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.

Isolate the Variable
In the first step of solving the inequality \(-2y > -7\), your primary goal is to isolate the variable, which in this case is \(y\). Isolating the variable means getting \(y\) by itself on one side of the inequality. This is done by performing inverse operations until \(y\) is left alone.

  • In our problem, we start by dividing both sides of the inequality by -2, because -2 is the coefficient of \(y\).
\[-2y > -7\] \[\frac{-2y}{-2} < \frac{-7}{-2}\]

Dividing both sides by -2 helps in reducing the inequality to: \y < \frac{7}{2}\.

Notice how isolating the variable sets us up to more easily understand the relationship it has with the numbers on the other side of the inequality.
Reverse Inequality Sign
When you divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This is an essential rule that protects the inequality's true meaning.

For instance, consider our example where we have the inequality: \ \frac{-2y}{-2} > \frac{-7}{-2}\. Normally, dividing or multiplying both sides of an equation by a positive number does not alter the inequality sign. But with negative numbers, it's different.
  • Think of it this way: When multiplying or dividing by a negative, you are essentially flipping the relationship between the numbers.

Therefore, after dividing the negative number from both sides, we change the \(>\) sign to a \(<\) sign, leading us to: \ y < \frac{7}{2} \
Dividing by Negative Number
When solving inequalities, dividing by a negative number is a crucial step that comes with a special rule. In our example, we had to divide by -2 to isolate \(y\). However, whenever you divide by a negative number, you must switch the direction of the inequality sign.

  • This step is often confusing, but it is an important rule to understand thoroughly. Ignoring it can result in an incorrect solution.

Here, we took \-2y > -7\ and divided by -2 resulting in: \y < \frac{7}{2}\

By understanding and mastering this rule, you will be better equipped to handle a wide range of inequalities involving negative numbers.

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

A person can drive from town A to town B at a certain rate of speed in 5 hours. If he increases his speed by \(15 \mathrm{kph}\), he can make the trip in 4 hours. How far is it from town A to town B?

Perform the indicated operations on the given inequality. Sketch the resulting inequality on a number line. \(-5<-x<-1 ;\) multiply each member by \(-1\)

Perform the indicated operations and simplify as completely as possible. $$3 a b\left(a^{2}-5 b\right)-3 b\left(a^{3}+4 a b\right)$$

Set up an inequality and solve it. Be sure to clearly label what the variable represents. An organization wants to sell tickets to a concert. It plans on selling 300 reserved-seat tickets and 150 tickets at the door. The price of a reserved- seat ticket is to be \(\$ 2\) more than a ticket at the door. If the organization wants to collect at least \(\$ 3,750\), what is the minimum price it can charge for a reserved-seat ticket?

Solve the inequality and sketch the solution set on a number line. $$x+4<2 x-1$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Probability and 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.