Problem 25 - Elementary and Intermediate Algebra

Question

Find \(-x\) when \(x\) is each of the following. $$ -\frac{14}{3} $$

Step-by-Step Solution

Answer

\( -(-\frac{14}{3}) = \frac{14}{3} \)

Step 1 - Understand the Problem

The task is to find the negative of the given value of \( x \). Here, \( x = -\frac{14}{3} \). We need to find \( -x \).

Step 2 - Apply the Negative Sign

To find \( -x \), multiply the given \( x \) by -1. So, we calculate: \[ -(-\frac{14}{3}) \]

Step 3 - Simplify the Expression

When you multiply two negative numbers, the result is positive. Therefore, \[ -(-\frac{14}{3}) = \frac{14}{3} \]

Key Concepts

Multiplying Negatives

Multiplying negative numbers can be tricky at first, but it's quite simple once you get the hang of it. The rule is: A negative number multiplied by another negative number results in a positive number. This is essential in algebra, especially when simplifying expressions. Consider the exercise given, where we have to find \(-x\) if \(x = -\frac{14}{3}\). By multiplying \x\ by \-1\ (because \(-x) means \x\ times \-1\), we get: \-(-\frac{14}{3})\. Since both numbers are negative, the product turns positive, giving us \(\frac{14}{3}\). This property helps simplify and solve many algebraic expressions, making it easier to work with complex problems.

Integer Operations

Integer operations involve adding, subtracting, multiplying, and dividing whole numbers. It's crucial to understand these basics for algebra. In the exercise, we used multiplication of integers. Here are some core rules to remember when dealing with integer operations:

• Adding two positive integers always gives a positive result.
• Adding two negative integers results in a sum that's more negative.
• Multiplying or dividing two integers with the same sign (both positive or both negative) results in a positive number.
• Multiplying or dividing two integers with different signs results in a negative number.

These rules help in performing operations on negative numbers, simplifying expressions, and solving equations. Practice frequently to become comfortable with these operations.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is about making them easier to work with. It involves performing operations and reducing the expression to its simplest form. Let's break it down:

• **Combine like terms:** Terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients.
• **Apply the distributive property:** This helps to eliminate parentheses. For instance, \(a(b + c) = ab + ac\).
• **Handle negative signs:** As seen in our exercise, know how to multiply and divide by negative numbers correctly to simplify the expression.

Always perform operations step-by-step to avoid errors. Simplifying expressions makes it easier to solve equations and understand the relationship between variables. Practice makes perfect, so keep working on these concepts regularly.