Problem 33 - Elementary Algebra

Question

Determine each of the values, \(-1-(-8) \mid\)

Step-by-Step Solution

Answer

Answer: 7

Step 1: Handle parentheses and subtraction inside#

We'll first deal with the parentheses and compute the subtraction inside them: \(-1 - (-8)\). Since we are subtracting a negative number, it is equivalent to adding the positive version of that number: \(-1 + 8\).

Step 2: Perform the addition#

Compute the sum: \(-1 + 8 = 7\).

Step 3: Compute the absolute value#

Now, compute the absolute value of the result: \(|7|\).

Step 4: Final result#

The absolute value of 7 is 7, so the final result is: \(|-1-(-8) \mid = 7\).

Key Concepts

Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is always a non-negative value. For example, the absolute value of both \( -3 \) and \( 3 \) is \( 3 \). When solving absolute value equations, it's important to recognize that it's possible to have two different numbers with the same absolute value, one positive and one negative.

To find the absolute value of a number or an algebraic expression, you use the notation \( |x| \), where \( x \) is the number or expression in question. If \( x \) is negative, \( |x| \) equals \( -x \), which makes it positive. If \( x \) is already positive, \( |x| \) equals \( x \) itself. For our exercise, the absolute value of \( 7 \) is simply \( 7 \), because it's already positive.

Operations with Integers

When working with integers, understanding how to perform basic operations such as addition, subtraction, multiplication, and division is crucial. In our example, we encounter addition and subtraction of integers. Remember, subtracting a negative is the same as adding its positive counterpart. Hence, \( -1 - (-8) \) becomes \( -1 + 8 \).

Here are some simple rules to follow when performing operations with integers:

• Addition of two positive integers gives a positive result.
• Addition of two negative integers gives a negative result.
• Subtracting a larger number from a smaller one switches the sign.
• Multiplication and division follow the rule: Positive times positive or negative times negative gives a positive result, while positive times negative gives a negative result.

In this case, we add a negative and a positive number. Since the positive number is greater, the result is positive.

Parentheses in Algebra

Parentheses in algebra serve to organize the order of operations and to signify that the operations enclosed within them should be performed first. They are also used to indicate multiplication or to clarify operations being performed on negative numbers.

In our example, the expression \( -1-(-8) \) includes a parentheses around the \( -8 \) to indicate that we are subtracting the entire value of \( -8 \) from \( -1 \). If a number is inside the parentheses with a minus sign directly outside, as in this example, it means you are subtracting a negative, which is equivalent to adding the positive of that number. Always resolve operations inside the parentheses as the initial step.

Tip for Parentheses

If you ever see \( -(x) \) where \( x \) is a positive number, you can remove the parentheses by simply writing it as \( -x \).