Problem 8 Solve the inequality. $$ 4 x... [FREE SOLUTION]

Chapter 6: Problem 8

Solve the inequality. $$ 4 x-1>7 \text { or } 5 x-1<-6 $$

Short Answer

Expert verified

The solution to the inequality is $x > 2$ or $x < -1$.

Step by step solution

Solving the first inequality: $4x - 1 > 7$

Starting with the first inequality, $4x - 1 > 7$, we isolate x by first adding 1 to both sides of the inequality which results in $4x > 8$. Then, we divide both sides of the inequality by 4 which gives us $x > 2$.

Solving the second inequality: $5x - 1 < -6$

Similarly, we start solving the second inequality, $5x - 1 < -6$, again by adding 1 to both sides of the inequality, resulting in $5x < -5$. Then, we divide both sides of the inequality by 5, which gives us $x < -1$.

Union of solutions

Since the inequalities are connected by the operator 'or', the solution is the union of the solutions of the two inequalities. Therefore, x can be any value greater than 2 or any value less than -1.

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.

Solving Inequalities

Inequalities are expressions that use symbols like '>' (greater than), '<' (less than), '鈮�' (greater than or equal to), and '鈮�' (less than or equal to) to show the relationship between two values. When solving inequalities, our goal is to find the values of a variable that make the inequality true. The process is similar to solving equations but with an added rule regarding multiplication or division by a negative number.
When you multiply or divide an inequality by a negative number, you need to flip the inequality sign. For example, if you multiply both sides of an inequality by -1, a '>' would become a '<'. Let's apply these steps to our exercise:

Compound Inequalities

Compound inequalities involve more than one inequality joined together by the words 'and' or 'or'. Each word affects how we solve them.

'And' means both conditions must be true; solutions are found in the overlap of the inequalities.
'Or' indicates that if any part of the inequality is true, the solution is valid.

In this exercise, our compound inequality is connected by the word 'or'. This means we need to find solutions that satisfy either one inequality or the other, or even both in some cases. The solutions to this compound inequality will be the union, which includes values from x > 2 or x < -1. These solutions cover separate ranges on the number line.

Inequality Solutions

The possible solutions for inequalities can often be expressed on a number line. This gives a visual representation of which values satisfy the inequality. For example, for 'x > 2', you would shade all numbers greater than 2. For 'x < -1', shade all numbers less than -1. With compound inequalities that use 'or', the solution includes all numbers covered by both shaded areas.
It鈥檚 important to remember:

Use an open circle to denote numbers that are not included in the solution set (e.g., greater than or less than).
Use a closed circle for numbers that are included (e.g., greater than or equal to).

The final solution will allow any value for x that lies outside the interval from -1 to 2. This includes all values less than -1 and all values greater than 2.

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

Solve the inequality. $$ 4 x-1>7 \text { or } 5 x-1<-6 $$

Short Answer

Step by step solution

Solving the first inequality: \(4x - 1 > 7\)

Solving the second inequality: \(5x - 1 < -6\)

Union of solutions

Key Concepts

Solving Inequalities

Compound Inequalities

Inequality Solutions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Calculus

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.