Problem 55 $$ \text { For Problems 23-60,... [FREE SOLUTION]

91影视

Elementary Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 3: Problem 55

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 4 x+2-6 x<-1 $$

Short Answer

Expert verified

x > 1.5

Step by step solution

Combine Like Terms

Start solving the inequality by simplifying and combining the like terms on the left side. The inequality is given as: \[4x + 2 - 6x < -1\] Combine the terms with $x$: \[(4x - 6x) + 2 < -1\] This simplifies to: \[-2x + 2 < -1\]

Isolate the Variable Term

To isolate the variable term, subtract 2 from both sides of the inequality: \[-2x + 2 - 2 < -1 - 2\] This simplifies to: \[-2x < -3\]

Solve for x

To solve for $x$, divide both sides by -2. Remember, dividing by a negative number reverses the inequality sign: \[x > \frac{-3}{-2}\] This simplifies to: \[x > \frac{3}{2}\]

Write the Final Answer

The solution to the inequality is: \[x > \frac{3}{2}\] This means $x$ must be greater than 1.5.

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.

Combining Like Terms

When you have an inequality or equation, one of the first steps to simplify it is by combining like terms. Like terms are terms that have the same variables raised to the same powers. They can be combined by adding or subtracting their coefficients.
In our original inequality, \[4x + 2 - 6x < -1\]the terms $4x$ and $-6x$ are like terms because they both involve the variable $x$.

Combine $4x$ and $-6x$ by performing the operation $4 - 6$, which equals $-2$.
Replace $4x - 6x$ with $-2x$ in the inequality.

This gives us the simplified inequality:
\[-2x + 2 < -1\]Combining like terms is essential to simplify equations and inequalities, making them easier to solve.

Isolating the Variable

Isolating the variable is a crucial step in solving inequalities or equations. The goal is to get the variable on one side by itself. This makes it easier to see what values satisfy the inequality.
For our inequality $-2x + 2 < -1$, begin by removing the constant term from the side with the variable term.

Subtract 2 from both sides of the inequality: $-2x + 2 - 2 < -1 - 2$
This simplifies to: $-2x < -3$

By subtracting, we successfully isolate the variable term, $-2x$, on one side.
This step moves us closer to finding the solution for $x$. Keep in mind, when you move terms around in an inequality, what you do to one side must be done to the other to maintain balance.

Reversing Inequality Sign

Reversing the inequality sign is a unique step that only occurs under specific circumstances when solving inequalities. This step is necessary when you multiply or divide both sides of an inequality by a negative number.
In our example, we have the inequality:\[-2x < -3\]To solve for $x$, divide both sides by $-2$.

Since dividing by a negative number, the inequality sign reverses: $x > \frac{-3}{-2}$
This simplifies to $x > \frac{3}{2}$

Remembering to reverse the inequality sign is critical to obtaining the correct solution. It transforms our inequality, maintaining the truth of the statement.
This transformation shows us clearly that $x$ must be greater than $1.5$. Always double-check your steps to ensure you haven't missed reversing the sign when necessary.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

$$ \text { For Problems 23-60, solve each inequality. (Objective 1) } $$ $$ 4 x+2-6 x<-1 $$

Short Answer

Step by step solution

Combine Like Terms

Isolate the Variable Term

Solve for x

Write the Final Answer

Key Concepts

Combining Like Terms

Isolating the Variable

Reversing Inequality Sign

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Discrete Mathematics

Decision Maths

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.