Problem 57 - Prealgebra

Question

Apply the distributive property, then simplify. $$\frac{2}{3}(-3 x-6)$$

Step-by-Step Solution

Answer

The expression simplifies to \(-2x - 4\).

Step 1: Identify the Distributive Property

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). Here, we have a difference \(-3x - 6\). We will distribute \(\frac{2}{3}\) to both \(-3x\) and \(-6\).

Step 2: Distribute to the First Term

Multiply \(\frac{2}{3}\) by \(-3x\). This is calculated as follows: \(\frac{2}{3} \times (-3x) = -2x\).

Step 3: Distribute to the Second Term

Multiply \(\frac{2}{3}\) by \(-6\). This is calculated as follows: \(\frac{2}{3} \times (-6) = -4\).

Step 4: Combine the Results

Write the expression as the sum of the results from steps 2 and 3. This results in \(-2x - 4\).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are like a language, communicating mathematical ideas through symbols and numbers. In our exercise, we have the expression \(\frac{2}{3}(-3x - 6)\).

• Here, \(-3x - 6\) combines a variable term \(-3x\) and a constant term \(-6\).
• The variable \(x\) can take different values. Consequently, the value of the entire expression changes with \(x\).
• Algebraic expressions are simplified using rules like the distributive property and combining like terms.

Algebraic expressions allow us to solve problems where numbers alone are insufficient. They offer flexibility and dynamism, as they can represent a range of scenarios.

Making Simplification Simple

Simplification aims to make expressions easier to work with by reducing their complexity. For instance, reducing \(\frac{2}{3}(-3x - 6)\) results in \(-2x - 4\). Why simplify?

• Focuses on reducing clutter and confusion by removing unnecessary elements.
• Finds a concise way to express the same mathematical idea.

To simplify:

• Identify and apply mathematical rules, like the distributive property, effectively.
• Combine like terms whenever possible.

Simplifying isn't just about shorter expressions. It's about gaining clarity and ensuring calculations are manageable.

The Art of Multiplying Fractions

Multiplying fractions seems tricky, but with a clear method, it's straightforward. Here's how we tackled multiplying \(\frac{2}{3}\) by each term separately in the expression.

• Multiply the numerators together (top numbers) and the denominators together (bottom numbers).
• Simplify the resulting fraction if possible.

In our solution:

• For \(\frac{2}{3} \times (-3x)\), multiply \(2\) by \(-3x\) and divide by \(3\), simplifying to \(-2x\).
• For \(\frac{2}{3} \times (-6)\), multiply \(2\) by \(-6\) and divide by \(3\), resulting in \(-4\).

Remember: fraction multiplication doesn't need common denominators, and it's all about straight multiplication and simplifying the results.