Problem 24 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{7}{2 x-3}-3\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-6x + 16}{2x-3} \).

Step 1: Identify the expression

The given expression is \( \frac{7}{2x-3} - 3 \). Our task is to simplify this expression.

Step 2: Rewrite the constant as a fraction

To simplify the subtraction, rewrite the constant \( 3 \) as a fraction with the same denominator as the first term: \( \frac{3(2x-3)}{2x-3} = \frac{6x-9}{2x-3} \).

Step 3: Subtract the fractions

Perform the subtraction of the fractions by subtracting their numerators: \( \frac{7 - (6x-9)}{2x-3} \).

Step 4: Simplify the numerator

Distribute the negative sign and simplify the numerator: \( 7 - 6x + 9 = -6x + 16 \).

Step 5: Write the final simplified expression

The simplified form of the expression is \( \frac{-6x + 16}{2x-3} \).

Key Concepts

Fraction Subtraction

When dealing with algebraic fraction subtraction, understanding how to handle different denominators is key. In exercises like \[ \frac{7}{2x-3} - 3 \] it's vital to transform each term to have a common denominator. First, convert the constant term into a fraction. Since 3 is a whole number, rewrite it as a fraction with the same denominator as the algebraic fraction:

• Identify the denominator of the fraction, in this case, \(2x-3\).
• Express 3 as \(\frac{3(2x-3)}{2x-3}\), which gives you the equivalent fraction \(\frac{6x-9}{2x-3}\).

With both terms now sharing a common denominator, subtract the fractions by subtracting their numerators. This technique ensures a smooth operation without altering the denominator itself. Remember: when subtracting fractions with the same denominator, focus only on the numerators.

Simplifying Expressions

Simplifying expressions, especially those involving fractions, requires careful handling of operations, including distributing signs and combining like terms. After setting up the expression:\[ \frac{7 - (6x-9)}{2x-3} \]Begin by addressing the subtraction in the numerator. Distribute the negative sign across the terms in the parentheses:

• Change \(- (6x - 9)\) to \(-6x + 9\).
• Add this to 7, giving the new numerator: \(7 - 6x + 9\).
• Simplify by combining like terms: \(16 - 6x\).

This simplification process helps to make algebraic expressions more manageable, particularly when the goal is to further apply operations like evaluating or graphing the expression. Always double-check each step to ensure accuracy.

Numerical Operations

Numerical operations in algebra involve a variety of basic math principles, such as addition, subtraction, and distribution, that are crucial when handling complex expressions. In solving \[ \frac{7 - (6x-9)}{2x-3} \]addition and subtraction processes follow logical steps. Addition combines values while subtraction separates them. For instance:

• In \(7 - (6x - 9)\), transform to \(7 - 6x + 9\).
• Combine the numerals: \(7 + 9 = 16\).
• Arrive at \(-6x + 16\) by following these basic arithmetic rules.

Always apply mathematical operations carefully, considering their priority: parentheses first, followed by exponents, multiplication, division, and finally addition and subtraction (PEMDAS). Keeping these fundamentals in mind ensures effective problem-solving in algebra.