Problem 23 - Elementary and Intermediate Algebra

Question

Find \(-x\) when \(x\) is each of the following. $$ -45 $$

Step-by-Step Solution

Answer

-x = 45 when x = -45

Step 1: Understand the Problem

The problem requires finding \(-x\) when \( x = -45 \). This means we need to determine the opposite value of \( x \).

Step 2: Apply the Formula

To find \(-x\), multiply \( x \) by -1. So, \[ -x = -(-45) \].

Step 3: Simplify the Expression

Simplifying \( -(-45) \), we get positive 45 since multiplying two negative numbers results in a positive number. Thus, \( -(-45) = 45 \).

Key Concepts

Multiplication

Multiplication is one of the basic operations in mathematics. It combines two numbers to give a product. In this problem, we use multiplication to find the opposite value of a number. It is important to know that when we multiply two negative numbers, the product is always positive. This is because the negative signs cancel each other out. For example, if we have \( x = -45 \), to find the opposite value \( -x \), we multiply \( x \) by -1. So, \[ -x = -1 \times (-45) \]. This results in \(+45\).
In this context:

• Multiplying two positive numbers gives a positive product.
• Multiplying one positive and one negative number gives a negative product.
• Multiplying two negative numbers gives a positive product.

Opposite Value

Finding the opposite value of a number means finding the number that, when added to the original, results in zero. Essentially, it is the negative version of that number. In the given problem, we are asked to find the opposite value of \( x = -45 \). To do this, we multiply \( x \) by -1.
The important part to note is: the opposite value of a negative number is positive.
For instance:

• The opposite of \(-45\) is \(+45\).
• The opposite of \( -10 \) is \( +10 \).

When two negative signs are multiplied, they result in a positive sign. Thus, \[ -(-45) = 45 \].

Simplification

Simplification refers to the process of reducing an expression or equation to its simplest form. In the given problem, we start with the expression \[ -(-45) \]. To simplify, we need to remember the rule for multiplying two negative numbers. The product of two negative numbers is positive, so \[ -(-45) = 45 \].
That means we're essentially canceling out the negative signs.
This simplification can be broken down into:

• Identifying the number and its opposite.
• Applying the multiplication rule for negative numbers.
• Writing the final simplified positive value.

Understanding simplification is crucial for solving many mathematical problems. In this case, it shows that \(-(-45)\) yields a neat result of \(45\).