Problem 12 - Elementary and Intermediate Algebra

Question

Complete each solution. $$ \text { True or false: } 6 x+4+\frac{-3}{x+2}=6 x+4-\frac{3}{x+2} $$

Step-by-Step Solution

Answer

True, both sides are identical after simplification.

Step 1: Analyze the Equation

First, observe the given statement: \(6x + 4 + \frac{-3}{x+2} = 6x + 4 - \frac{3}{x+2}\). We want to determine if it is true or false by comparing both sides of the equation.

Step 2: Simplifying the Left Side

Simplify the left side of the equation. It is written as \(6x + 4 + \frac{-3}{x+2}\), which is equivalent to \(6x + 4 - \frac{3}{x+2}\) because adding a negative fraction is the same as subtracting the positive fraction.

Step 3: Simplify Both Sides

Both sides of the equation are now: \(6x + 4 - \frac{3}{x+2}\). They are identical, meaning both expressions are equal.

Step 4: Conclusion and Verification

Since both sides of the equation are identical after simplification, the statement is true. Always verify each side simplifies to the same expression.

Key Concepts

Equation Simplification

Equation simplification is a key step in determining if two algebraic expressions are equal. We simplify equations by performing basic arithmetic operations, such as addition and subtraction, to make the equation more manageable and often easier to understand. In our exercise, simplification occurs when we rewrite the left side of the equation

• The left side starts as \(6x + 4 + \frac{-3}{x+2}\)
• We simplify by recognizing that adding a negative is the same as subtracting a positive, which changes it to \(6x + 4 - \frac{3}{x+2}\).

This transformation is important because it aligns the expressions on either side, allowing us to easily see they are equivalent by simplifying the negative fraction to subtraction. With practice, simplifying equations becomes a smooth and intuitive part of working with algebra.

Fractions in Algebra

Fractions with algebraic terms can initially appear complex, but they follow the same principles as numerical fractions. Understanding these fundamentals can simplify the process greatly. An algebraic fraction may include variables in the numerator, denominator, or both.

• In the given exercise, there was a term \(\frac{-3}{x+2}\) on the left side, which can be rewritten as \(-\frac{3}{x+2}\).
• This is crucial as it demonstrates how fractions interact with the rest of the algebraic expression.

The fraction here involves a variable \(x+2\) in the denominator. When comparing such fractions in equations, ensure the operations adhere to standard rules of arithmetic, ensuring consistency across both sides of the equation. Recognizing these patterns makes algebraic fractions less daunting and more approachable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Mastering these expressions is crucial in algebra. They form the foundation of more complex algebraic processes, like solving equations or factoring. In the equation provided:

• The expression \(6x + 4 + \frac{-3}{x+2}\) was restructured using basic algebraic operations to become \(6x + 4 - \frac{3}{x+2}\).
• This demonstrates that algebraic expressions are flexible and can often be rewritten or rearranged to reveal their true equality.

Thinking about algebraic expressions in terms of rearrangement and simplification can assist in solving complex algebra problems. Familiarity with expressions aids in understanding the overall behavior of equations, ensuring a smoother problem-solving process.