Problem 96 - Beginning and Intermediate Algebra

Question

If \(a\) is a positive number and \(b\) is a negative number, fill in the blanks with the words positive or negative. \(b+b\) is

Step-by-Step Solution

Answer

\( b + b \) is negative.

Step 1: Understanding the Problem

We are given that \( a \) is a positive number and \( b \) is a negative number. We need to determine whether the sum \( b + b \) is positive or negative, using the information that \( b \) is negative.

Step 2: Operation on Negative Numbers

Recall that adding two negative numbers will result in another negative number. This is because when you add a negative value to a negative value, you are essentially decreasing the overall value further, keeping it negative.

Step 3: Apply to the Given Problem

Apply the rule from the previous step specifically to \( b + b \). Since \( b \) is negative, \( b + b \) is the same as adding the negative quantity \( b \) to itself. The result of \( b + b \) will still be negative.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero and are denoted with a minus sign (-). They play an essential role in mathematics, especially when dealing with real-world situations, like temperature below zero or debts.
Negative numbers follow specific rules when it comes to operations:

• When you add two negative numbers, the result is still negative. For example, adding \(-5 + (-3) = -8\).
• Multiplying two negative numbers results in a positive number.
• When you subtract a negative number, it's the same as adding the positive equivalent of that number. For instance, \(-6 - (-3) = -6 + 3 = -3\).

Understanding these basic operations with negative numbers can significantly help when solving algebraic problems.

Addition of Integers

In mathematics, integers include all whole numbers and their negative counterparts. Working with integers involves understanding how to add them, whether they are positive or negative.
Here are some rules to remember when adding integers:

• Adding two positive integers always results in a positive integer. For example, \(3 + 4 = 7\).
• Adding two negative integers, as in the problem we're considering, results in a negative integer as demonstrated with \(-5 + (-5) = -10\).
• When adding a positive integer and a negative integer together, the result depends on the magnitude (size) of the integers. If the positive integer is larger, the result is positive, and if the negative integer is larger, the result is negative.

Remember these guidelines whenever working with integer addition to simplify and solve expressions correctly.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are used to represent real-world situations and solve problems systematically.
For example, in the problem provided, \(b + b\) is an algebraic expression where, rather than using numerical values, variables represent numbers.Here are some key points about algebraic expressions:

• Variables like \(x\) or \(b\) represent unspecified or unknown numbers.
• Operations such as addition, subtraction, multiplication, and division are used to combine these variables in expressions.
• Simplifying algebraic expressions involves combining like terms and making them easier to work with.
• The expressions can be evaluated by substituting variables with actual numbers.

Understanding how to interpret and manipulate algebraic expressions is crucial in solving mathematical problems efficiently.