Problem 36 - Intermediate Algebra

Question

Solve equation. \(\frac{1}{2 d-4}-\frac{1}{2-d}=\frac{1}{4}\)

Step-by-Step Solution

Answer

The solution is \(d = 8\).

Step 1: Rearrange the Equation

Let's first rewrite the equation with a common denominator for the left-hand side to simplify the terms. The equation is: \[ \frac{1}{2d-4} - \frac{1}{2-d} = \frac{1}{4} \] To find a common denominator for the left-hand side fractions, notice that \(2d-4\) can be rewritten as \(2(d-2)\), and \(2-d\) can be rewritten as \(-(d-2)\). Thus, the common denominator is \(2(d-2)\).

Step 2: Simplify the Fraction Expression

Rewrite each fraction with the common denominator: \[ \frac{1}{2(d-2)} + \frac{1}{(d-2)} = \frac{1}{4} \] The negative sign from \(-(d-2)\) has been absorbed in the second fraction making it an addition problem. Convert the fractions: \[ \frac{1}{2(d-2)} + \frac{2}{2(d-2)} = \frac{1}{4} \] The numerator of the second fraction was multiplied by 2 to match the denominator. Combine the fractions: \[ \frac{1 + 2}{2(d-2)} = \frac{1}{4} \] Simplifying gives: \[ \frac{3}{2(d-2)} = \frac{1}{4} \]

Step 3: Cross-Multiply to Solve

To eliminate the fractions, cross-multiply: \[ 3 imes 4 = 2(d-2) \times 1 \] This simplifies to: \[ 12 = 2(d-2) \]

Step 4: Solve for d

Now, distribute and solve for \(d\):\[ 12 = 2d - 4 \] Add 4 to both sides:\[ 16 = 2d \] Divide both sides by 2:\[ d = 8 \] Thus, the solution to the equation is \(d = 8\).

Key Concepts

Understanding Common Denominators

When dealing with rational equations, a common denominator enables us to combine fractions easily. This ensures the fractions have equal-sized parts, making operations such as addition and subtraction straightforward. Notice in the original expression: \( \frac{1}{2d-4} - \frac{1}{2-d} \). Before we can solve, we need consistency. Here's why:

• The denominator \(2d-4\) is equivalent to \(2(d-2)\).
• The denominator \(2-d\) is equivalent to \(-(d-2)\), which flips the sign of the fraction.

Thus, a common denominator for these is \(2(d-2)\). By expressing the fractions with this common base, we align their denominators, simplifying further calculations. Think of it as putting two differently sized pieces into a puzzle: standardizing their size lets them fit together seamlessly.

The Strategy of Cross-Multiplication

Cross-multiplication is a useful technique for solving equations involving two fractions. It simplifies the process by eliminating fractions from the equation, allowing us to focus on solving the variable directly. Here's how it applies:The simplified fraction we developed was: \( \frac{3}{2(d-2)} = \frac{1}{4} \) Steps:

• Multiply the numerator of the first fraction (3) by the denominator of the second fraction (4): \( 3 \times 4 \).
• Do the same with the denominator of the first fraction and the numerator of the second fraction: \( 2(d-2) \times 1 \).

Thus, this gives us: \[ 12 = 2(d-2) \] This is simpler to solve than the fraction-fraught previous step. By cross-multiplying, we've converted into a form that's visibly straightforward to manipulate in finding \(d\). This is why cross-multiplication is so popular—it delivers clarity from fractions.

The Steps of Solving Equations

Once an equation is free from fractions via simplifications like cross-multiplication, the path to solving it is clearer. Let's explore these crucial steps:Take the equation we got: \( 12 = 2(d-2) \) Here's how to solve:

• Distribute the 2 into the parenthesis: \( 2d - 4 \).
• Align both sides by adding 4, removing constants: \( 16 = 2d \).
• Divide each side by 2 to isolate \(d\): \( d = 8 \).

Congratulations, you've solved for \(d\)! Each action—distributing, aligning, and isolating—is crucial. These are core to transforming equations into simple statements about the variable. Remember, consistency and diligence at each stage make solving any equation systematic and trouble-free.