Problem 90 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ (1-8)-(2-9) $$

Step-by-Step Solution

Answer

0

Step 1 - Simplify inside the parentheses

Start by simplifying each expression inside the parentheses: (1-8) = -7 (2-9) = -7

Step 2 - Substitute the simplified values

Replace the original expressions with their simplified values: (1-8)-(2-9) becomes -7 - (-7)

Step 3 - Simplify the resulting expression

Next, simplify the expression by subtracting: -7 - (-7) is the same as -7 + 7, which equals 0

Key Concepts

parentheses simplification

Simplifying algebraic expressions often starts by dealing with the innermost parentheses first. They act as a signal to perform operations inside them before tackling the rest of the expression. Let's look at our example: \((1-8)-(2-9)\)
To simplify, focus on the expressions inside the parentheses first:
• \((1-8)\) and \((2-9)\).
Work out each part individually.
• \(1-8\) simplifies to \(-7\).
• \(2-9\) simplifies to \(-7\).
By handling what's inside the parentheses first, we ensure an orderly approach to more complex expressions.

subtraction of integers

Subtraction of integers is a key operation in algebra. When you subtract one number from another, you are essentially adding its opposite. Consider our simplified example: \(-7 - (-7)\)
To understand this, remember: subtracting a negative integer is the same as adding its positive counterpart. So, \(-7 - (-7)\) becomes \(-7 + 7\).
This process can be visually aided by thinking in terms of a number line. Moving to the left for negative numbers and right for positive ones.
• Start at \(-7\).
• Then move 7 steps to the right (adding 7).
This lands you at 0.

negative numbers

Understanding negative numbers is crucial in algebra. They represent values less than zero and have unique properties when added, subtracted, multiplied, or divided. In our example, both numbers inside parentheses resulted in negatives: \(-7\)
When working with negative numbers:

• Adding two negative numbers results in a more negative number. For instance, \(-2 + -3 = -5\).
• Subtracting a negative number from another is like adding the positive counterpart (as seen with \(-7 - (-7)\)).
• Multiplying and dividing negative numbers follow distinct rules: \ \begin{itemize} \item Negative * Negative = Positive.\ \item Negative * Positive = Negative. \item Negative \/ Positive = Negative. \item Negative \/ Negative = Positive.\ \end{itemize}

Grasping these concepts allows you to simplify and solve algebraic expressions with confidence.