Problem 87 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(\frac{1}{2}(2 x+4)-\frac{1}{3}(9 x-6)\)

Step-by-Step Solution

Answer

-2x + 4

Step 1: Distribute the Constants

Distribute \( \frac{1}{2} \) and \( \frac{1}{3} \) through their respective parentheses: \( \frac{1}{2}(2x + 4) - \frac{1}{3}(9x - 6) = \frac{1}{2} \cdot 2x + \frac{1}{2} \cdot 4 - \frac{1}{3} \cdot 9x + \frac{1}{3} \cdot 6 \).

Step 2: Simplify Each Term

Calculate the multiplication for each term: \( \frac{1}{2} \cdot 2x = x \, \frac{1}{2} \cdot 4 = 2 \, \frac{1}{3} \cdot 9x = 3x \ , \frac{1}{3} \cdot 6 = 2 \). Substitute back into the expression: \( x + 2 - 3x + 2 \).

Step 3: Combine Like Terms

Combine like terms to simplify the expression: \( (x - 3x) + (2 + 2) = -2x + 4 \).

Key Concepts

Distributive Property

When simplifying algebraic expressions, recognize the distributive property first. This property states that you can distribute a constant to each term within parentheses. In our exercise, we distribute \(\frac{1}{2}\) and \(\frac{1}{3}\) to the terms inside their respective parentheses. So, \[ \frac{1}{2}(2x+4)-\frac{1}{3}(9x-6)\] becomes \[ \frac{1}{2} \times 2x + \frac{1}{2} \times 4 - \frac{1}{3} \times 9x + \frac{1}{3} \times 6 \]. This step ensures each term inside the parentheses is multiplied by the constant outside.

Combining Like Terms

Once you've distributed the constants, the next crucial step is to combine like terms. Like terms are terms that contain the same variable raised to the same power. In our case, after distributing, we have the expression \[ x + 2 - 3x + 2 \]. Here, \(x\) and \(-3x\) are like terms. Similarly, \(2\) and \(2\) are like terms.
To combine them, you add or subtract the coefficients. So, \(x - 3x = -2x\) and \(2 + 2 = 4\). This simplifies our expression to \(-2x + 4\). When combining like terms, always ensure you only combine terms that have exactly the same variables and exponents.

Algebraic Simplification

Algebraic simplification involves making an expression as simple as possible. Following the distributive property and combining like terms, our expression is already simplified to \(-2x + 4\). Simplification makes the expression easier to understand and use.
In general, algebraic simplification includes:

• Using the distributive property to remove parentheses
• Combining like terms to condense the expression
• Reducing fractions if applicable
• Rewriting the expression in a more compact, easily understood form

So, to sum up, always follow these steps to simplify: distribute the constants, combine like terms, and reduce where possible.