/*! 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 31 Solve each inequality. $$ (5... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 31

Solve each inequality. $$ (5 x-1)^{2} \geq 0 $$

Short Answer

Expert verified
The solution is \( x \in \mathbb{R}. \)

Step by step solution

01

Simplify the Expression

Observe that \( (5x - 1)^2 \) is the square of a binomial. Since the square of any real number is always non-negative, \( (5x - 1)^2 \) is always greater than or equal to 0.
02

Analyze the Inequality

Because \( (5x - 1)^2 \) is always greater than or equal to 0 for all real numbers, the inequality \( (5x - 1)^2 \geq 0 \) holds for all \( x \) in the set of real numbers, \( \mathbb{R} \).
03

Final Solution

Therefore, the solution to the inequality is all real numbers, or \( x \in \mathbb{R}. \)

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.

binomial expressions
Binomial expressions are algebraic expressions that consist of two terms separated by a plus or minus sign. In the inequality \( (5x - 1)^2 \geq 0 \), the term \( 5x - 1 \) is a binomial.

A binomial expression can be represented in the general form \( a + b \) or \( a - b \), where \( a \) and \( b \) are any algebraic terms. Here, \( a = 5x \) and \( b = 1 \).

When a binomial is squared, we use the formula \( (a \, \text{term} \, b)^2 \), which expands to \( a^2 \, \text{term} \, 2ab + b^2 \). For our binomial \(5x - 1\), the squared form is \((5x - 1)^2\). This squares both terms and multiplies twice the product of the terms: this always results in a non-negative value.
real numbers
Real numbers include all the numbers you can think of: positive, negative, and zero, and they can be expressed as fractions, decimals, or whole numbers. The set of real numbers is denoted by \( \mathbb{R} \).

In the given inequality \( (5x - 1)^2 \geq 0 \), the solution indicates that the inequality holds for all real numbers.

This means any value of \( x \) from the set of real numbers will satisfy the inequality. Simply put, no matter what real number \( x \) you substitute into the expression, the result will always be greater than or equal to zero because squares of real numbers are non-negative.
non-negative values
Non-negative values are numbers that are either greater than or equal to zero. They include all positive numbers and zero, but not negative numbers.

In the given inequality \( (5x - 1)^2 \geq 0 \), the expression \( (5x - 1)^2 \) is always non-negative because it is a square of a real number.
This means it does not matter what value of \( x \) you choose; the squared term will never be less than zero.

Even if \( x \) is such that \( 5x - 1 = 0 \), the square of zero is zero, which is a non-negative value. This understanding is crucial, as it is the reason why the solution to the inequality includes all real 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

The following exercises are not grouped by type. Solve each equation. $$2 x^{4}-9 x^{2}=-2$$

Two pipes together can fill a tank in \(2 \mathrm{hr}\). One of the pipes, used alone, takes \(3 \mathrm{hr}\) longer than the other to fill the tank. How long would each pipe take to fill the tank alone?

Solve each inequality, and graph the solution set. $$ \frac{x-8}{x-4}<3 $$

In the 1939 classic movie The Wizard of Oz, Ray Bolger's character, the Scarecrow, wants a brain. When the Wizard grants him his "Th.D." (Doctor of Thinkology), the Scarecrow replies with the following statement. Scarecrow: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. His statement sounds like the formula for the Pythagorean theorem. Use wording similar to that of the Scarecrow, but state the Pythagorean theorem correctly.

Solve each equation. Check the solutions. $$(x+3)^{2}+5(x+3)+6=0$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Calculus

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.