Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

State the real number property that iustifies the statement $$ \frac{(x+1)(x-3)}{(2 x+1)(x-3)}=\frac{x+1}{2 x+1} $$

Step-by-Step Solution

Answer

The real number property that justifies the given statement is the cancellation property. This property allows us to simplify the left-hand side of the equation by canceling the common factor \((x-3)\) from both the numerator and the denominator, resulting in the simplified equation \( \frac{x+1}{2x+1}\).

Step 1: 1. Identify the given statement

Consider the equation: \[ \frac{(x+1)(x-3)}{(2x+1)(x-3)}=\frac{x+1}{2x+1} \]

Step 2: 2. Analyze the LHS of the equation

The left-hand side of the equation is a fraction with a numerator containing the product of two factors \((x+1)\) and \((x-3)\), and a denominator containing the product of two factors \((2x+1)\) and \((x-3)\).

Step 3: 3. Identify a common factor on LHS

On the left-hand side of the equation, both the numerator and denominator have a common factor, \((x-3)\).

Step 4: 4. Simplify the LHS by applying the real number property

The real number property that allows us to simplify the left-hand side is the cancellation property. We can cancel the common factor \((x-3)\) from both the numerator and the denominator of the fraction: \[ \frac{(x+1)(x-3)}{(2x+1)(x-3)} \Longrightarrow \frac{x+1}{2x+1} \]

Step 5: 5. Conclusion

The real number property that justifies the given statement is the cancellation property. The cancellation property allows us to cancel out the common factor \((x-3)\) from both the numerator and the denominator of the fraction on the left-hand side, thus simplifying the left-hand side to the right-hand side: \[ \frac{(x+1)(x-3)}{(2x+1)(x-3)}=\frac{x+1}{2x+1} \]

Key Concepts

Simplification of Fractions

Simplifying fractions is a fundamental skill that helps make complex mathematical problems more manageable. Think of it as the process of reducing fractions to their simplest form, where the numerator and the denominator have no common factors other than 1. To simplify a fraction, you look for any common factors in both the numerator and the denominator and divide them out.

For instance, in the fraction \[ \frac{6}{8} \], both 6 and 8 can be divided by 2, which is a common factor. The simplified form will therefore be \[ \frac{3}{4} \]. Simplifying fractions can help you with further mathematical operations such as addition, subtraction, multiplication, and division of fractions.

Cancellation Property

The cancellation property is a handy tool that can simplify algebraic fractions, especially when you see the same factor in both the numerator and the denominator of a fraction. It states that if a factor is present on both sides of a fraction, it can be 'cancelled out', or in other words, be divided out of both sides.

This property is derived from the fact that any number divided by itself is 1, and multiplying by 1 has no effect on another number. For example, in the expression \[ \frac{ab}{ac} \], 'a' is common to both the numerator and the denominator, so it can be cancelled: \[ \frac{b}{c} \]. The cancellation property makes computations simpler and is a time-saver in algebra.

Common Factors

Common factors are numbers that divide evenly into two or more other numbers. When working with fractions, identifying common factors in the numerator and denominator is key to simplifying them efficiently. In algebra, factors can also include variables or algebraic expressions.

For example, the number 12 has factors 1, 2, 3, 4, 6, and 12. If you're simplifying the fraction \[ \frac{12}{24} \], you'd observe that both 12 and 24 can be divided by 12, which is a common factor. You would then simplify the fraction to \[ \frac{1}{2} \]. Recognizing common factors quickly becomes an invaluable skill for simplifying algebraic fractions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like x or y), and operation symbols (like addition or multiplication). Simplifying algebraic expressions often involves combining like terms (terms that have the same variables raised to the same power) and using various properties of real numbers, including distributive, associative, and commutative properties.

For example, the expression \( 2x + 3x \) can be simplified by combining the like terms of \( 2x \) and \( 3x \) to get \( 5x \). Simplifying expressions is an important step in solving algebraic equations and inequalities, as it can reveal the structure of an equation or make it easier to solve.