/*! 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 59 Decide whether the statement is ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 59

Decide whether the statement is true or false . If it is false, give a counterexample. The product \((-a) \cdot(-1)\) is always positive.

Short Answer

Expert verified
The statement is false. The counterexample is when a = 0. The product in this case is \( 0 \cdot -1 = 0 \), which is not positive.

Step by step solution

01

Understand the Statement

First, it's crucial to understand what the statement means. The statement is saying \( -a \cdot -1 \) is always positive. Let's check if it's true or false.
02

Test the Statement using an Example

Let's take \( a = 2 \). Now, \( -a = -2 \). Thus, substituting the value of \( -a \) in \( -a \cdot -1 \), we get \( -2 \cdot -1 = 2 \) which is indeed positive.
03

Generalize the Result

By introducing any number in place of \( a \), one can see that \( -a \cdot -1 = a \), which is the absolute value of \( a \) and is always positive when \( a \) is not equal to zero. However, if \( a = 0 \), then \( -a \cdot -1 = 0 \cdot -1 = 0 \), which is neither positive nor negative. Hence the statement is false.
04

Provide Counterexample

A counterexample is a case where the statement doesn't hold true. An example of this is when \( a = 0 \). In this case \( -a \cdot -1 = 0 \), which is not positive, thus making the statement false.

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.

Negative Numbers
Negative numbers can be a bit confusing, but with a little practice, they become much easier to handle. A negative number is essentially a number with a "minus" sign before it. This signifies that it is less than zero. For example, -3 is three units to the left of zero on the number line.
  • Negative numbers are used to represent values below zero, such as debts or temperatures below freezing.
  • They are opposite to positive numbers, which are greater than zero.
  • Zero is considered neutral, as it is neither positive nor negative.
When multiplying negative numbers, it鈥檚 important to remember that multiplying two negative numbers results in a positive number. This is because the two "negatives" effectively cancel each other out.
However, if you multiply a negative number by a positive number, the result will be negative.
Multiplication
Multiplication is a fundamental operation in mathematics that represents repeated addition. For example, multiplying 3 by 4 essentially means adding 3 four times: 3 + 3 + 3 + 3.
Multiplication has a few important properties:
  • Commutative Property: This means the order of multiplication doesn't change the result. For instance, 3 x 4 is the same as 4 x 3.
  • Associative Property: This allows numbers to be grouped in any manner. For example, (2 x 3) x 4 is the same as 2 x (3 x 4).
  • Distributive Property: This states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. For example, 2 x (3 + 4) is equal to (2 x 3) + (2 x 4).
Especially relevant in the context of multiplying negative numbers, consider that multiplying two negative values results in a positive, such as -2 x -1 = 2. This reflects the rule that a negative times a negative equals a positive.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations (addition, subtraction, multiplication, and division). Variables are letters that represent unknown numbers. In expressions like \(-a \cdot -1\), \(a\) is a variable that can take any value.
Here鈥檚 why algebraic expressions are important:
  • They allow us to generalize mathematical problems by using variables.
  • They help in solving equations and finding unknown values.
  • They provide a compact method to express complex mathematical relationships.
When handling algebraic expressions, especially with negative numbers, one must remember that:
- The negative sign affects the whole part of an expression it is placed in front of.
- Simplifying algebraic expressions often involves applying the basic rules of arithmetic, such as the distributive property.

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

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ 7 x-3 x(x+1) $$

LOGICAL REASONING Your friend does not understand how the product of \(a\) and \((b+c)\) is given by both \(a b+a c\) and \(b a+c a\) Use the properties of multiplication to write a convincing argument to show your friend that both statements are correct.

EVALUATING EXPRESSIONS Evaluate the expression. $$ 4-8-3 $$

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ 3 t(t-5)+6 t^{2} $$

FREIGHT TRAINS A train with 150 freight cars is used to haul two types of grain. Each freight car can haul 97.3 tons of barley or 114 tons of corn. Let \(n\) represent the number of freight cars containing corn. Which function correctly represents the total weight the train can haul? $$ \text { A. } W=97.3(150-n)+114 n \quad \text { B. } W=97.3 n+114(150-n) $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete 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.