Problem 40 - Algebra 1

Question

Simplify the variable expression. $$\frac{2}{3}\left(-\frac{3}{2} x\right)$$

Step-by-Step Solution

Answer

The simplified version of the variable expression is \( -x \)

Step 1: Analyze the expression

The expression can be broken down into two multiplications, \(\frac{2}{3} \) times \( -\frac{3}{2} \), and the outcome times \(x \).

Step 2: Multiply the fractions

When multiplying fractions, multiply the top numbers (the numerators), and do the same for the bottom numbers (the denominators). So, \(\frac{2}{3} \) times \(-\frac{3}{2} \) gives us \(-1 \).

Step 3: Multiply the result with \(x \)

The result, \( -1 \) times \(x \) gives us \( -x \).

Key Concepts

Multiplying Fractions

When multiplying fractions, the key is to remember that it's essentially about multiplying the numerators with one another, and then the denominators with one another. This process is straightforward and requires two main steps:

• First, take the numerators of each fraction and multiply them together. In our example, that would be the numerators 2 from \(\frac{2}{3}\) and -3 from \(-\frac{3}{2}\). This results in -6.
• Next, take the denominators of each fraction and multiply them together. For our fractions \(\frac{2}{3}\) and \(-\frac{3}{2}\), the denominators are 3 and 2, respectively. This gives us 6.

Once you have both results, combine them into a fraction. The fraction \(\frac{-6}{6}\) simplifies to -1. Pay attention to the signs, as they play a crucial role in arriving at the correct answer.

Variable Expressions

A variable expression includes numbers, variables (like \( x \)), and arithmetic operations. In the exercise provided, we have \(-\frac{3}{2} x\), where the variable \( x \) is part of a product with the fraction.
Variable expressions can be manipulated much like numerical expressions. However, they demand careful consideration of the variable itself. Variables are often placeholders representing unknown numbers, so any transformation applied to them should maintain the mathematical properties the variable might include.
In our example, after simplifying \(\frac{2}{3} \times -\frac{3}{2}\) to -1, the expression simplifies to \(-1 \times x\). Thus, the overall expression becomes \(-x\). This exercise exemplifies how fractional multipliers affect variable expressions.

Negative Numbers in Algebra

Understanding how negative numbers interact in algebra is vital, especially when multiplying or simplifying expressions. Let's consider two key points:

• Multiplying a positive number by a negative number results in a negative product. In algebra, this rule holds constant. For -3 and 2, multiplying them gives us -6.
• When an expression includes a negative factor, like \(-\frac{3}{2} x\), it transforms the sign of the entire multiplication. Therefore, even if the other multiplicand is positive (\(\frac{2}{3}\)), the outcome remains negative, resulting in \(-1 \times x = -x\).

Keep these rules in mind when handling negative numbers, as they can significantly change the direction of your calculations. Ensuring precision with signs is crucial, as mistakes can lead to misunderstandings and incorrect solutions.