Problem 93 - Beginning and Intermediate Algebra

Question

If \(a\) is a positive number and \(b\) is a negative number, determine whether each statement is true or false. \(b-a\) is always a negative number.

Step-by-Step Solution

Answer

True, \(b-a\) is always negative.

Step 1: Understanding the operation

The expression we are considering is \(b-a\). We know from the problem statement that \(a\) is a positive number and \(b\) is a negative number.

Step 2: Analyzing the sign of the expression

Since \(b\) is negative and \(a\) is positive, when you subtract \(a\) from \(b\), you are adding a positive number to a negative number, worsening the negativity. Hence, \(b-a\) would result in a more negative value than \(b\) alone.

Step 3: Conclusion

Since the subtraction of any positive number \(a\) (even if it's small) from a negative number \(b\) results in a larger negative number because you're essentially "adding" the negative parts, \(b - a\) must always be negative.

Key Concepts

Integer Operations

Understanding integer operations is essential when working with mathematical expressions. An integer is a whole number that can either be positive, negative, or zero. Operations involving integers include addition, subtraction, multiplication, and division.

When two integers are combined through these operations, their signs affect the outcome:

• Adding two positive integers always results in a positive integer.
• Adding two negative integers results in a more negative integer.
• Adding a positive integer to a negative one involves considering the larger absolute value; the result takes the sign of the larger number.
• Subtracting integers is similar to performing addition with a sign change. In essence, subtracting a number is the same as adding its opposite.

For example, subtracting a positive integer is akin to adding a negative integer, which is crucial in expressions such as \( b-a \). Understanding these operational rules aids significantly in solving problems involving integers.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are a key aspect of algebra and allow us to represent real-world problems mathematically. In the exercise, the expression \(b-a\) is an algebraic expression consisting of two variables, \(b\) and \(a\), and an operation of subtraction.

The behavior of an algebraic expression is often influenced by the nature and properties of the involved variables. In the given exercise, \(a\) represents a positive number, and \(b\) represents a negative number. Thus, the expression reflects the relationship between these values. Generally, understanding how varying these variables affects the outcome is vital.

• Variables in algebraic expressions can represent any number, often depicted as \(x\), \(y\), \(z\), etc.
• An algebraic expression can be simplified based on arithmetic operations rules.
• Identifying constant and variable terms helps in evaluating expressions effectively.

Working with algebraic expressions like \(b-a\) can help understand how combining different number types (positive and negative) affects the overall expression.

Subtraction of Numbers

Subtraction is a fundamental arithmetic operation that involves finding the difference between two numbers. It is essential to understand how subtraction works, especially with positive and negative numbers, to solve problems effectively.

When subtracting numbers:

• Subtracting a positive number from another number means you move to the left on the number line, making the result smaller.
• Subtracting a negative number is equivalent to adding a positive number, which moves you to the right on the number line, increasing the result.

Applying this to the expression \(b-a\), since \(b\) is negative and \(a\) is positive, the subtraction \(b-a\) becomes adding a negative incrementally worse. This means \(b\) plus the negative impact of minus \(a\) results in a negative value. Grasping this shift helps confirm that \(b-a\) will always yield a negative result in the given scenario.