Problem 19 - Elementary Algebra

Question

Determine each of the values, \(-|9|\)

Step-by-Step Solution

Answer

Answer: -9

Step 1: Find the absolute value of 9

To find the absolute value of any number, we should take the positive value of the number. So, the absolute value of 9 is simply: \(|9| = 9\)

Step 2: Negate the absolute value

Now, applying the negative sign in front of the absolute value we have: \(-|9| = -9\) So, the expression \(-|9|\) is equal to \(-9\).

Key Concepts

Negation

Negation is a mathematical operation that involves changing the sign of a number. Essentially, it means placing a negative sign in front of a number. In simple terms, if you have a positive number, negation will make it negative. Conversely, if you have a negative number, negation will make it positive. This is useful when working with mathematical expressions that require us to find the opposite value.

In the given exercise, after determining the absolute value of 9, which is 9, we apply negation. This means we transform 9 into -9 by adding the negative sign in front of it, giving us the final result of \(-|9| = -9\). Remember:

• If you negate a positive number, it becomes negative.
• If you negate a negative number, it becomes positive.

Integer Operations

Integer operations refer to basic arithmetic operations involving whole numbers, including addition, subtraction, multiplication, and division. Integers can be positive, negative, or zero. Understanding how these operations work with integers is crucial for solving various mathematical problems.

When dealing with integer operations, especially with negation, it’s important to keep track of positive and negative signs. In our example, working with the absolute value leads to handling the positive integer 9, which we then negate, resulting in -9. Here are some tips for working with integers:

• Adding two positive integers yields a positive sum.
• Adding two negative integers yields a negative sum.
• Subtracting a negative integer is equivalent to adding its positive counterpart.
• Multiplying or dividing two integers with the same sign results in a positive outcome, while doing so with different signs results in a negative outcome.

Arithmetic Expressions

Arithmetic expressions involve combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. These expressions can also include the operation of negation and the calculation of absolute values.

Consider the expression \(-|9|\) from the exercise. This is a simple arithmetic expression that involves finding the absolute value of an integer and then applying negation. Calculating absolute values involves finding the non-negative value of a given number, while negation involves changing the number's sign.

When evaluating arithmetic expressions, it's helpful to follow certain rules and order of operations. The rules generally follow the sequence of:

• Address any parentheses or absolute value operations first.
• Negate numbers as required by the expression.
• Perform multiplication and division from left to right.
• Complete any addition and subtraction last.

By understanding and applying these rules, evaluating even the more complex arithmetic expressions becomes straightforward.