Problem 34 - Intermediate Algebra

Question

Solve equation. \(\frac{1}{2 x-16}+\frac{14}{8-x}=\frac{3}{2}\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{11}{3}\).

Step 1: Identify the Common Denominator

To solve the equation \( \frac{1}{2x-16} + \frac{14}{8-x} = \frac{3}{2} \), we need to first eliminate the denominators by finding a common denominator. Notice that the denominators are \(2x-16\) and \(8-x\). We can rewrite \(2x-16\) as \(2(x-8)\); thus, the expression simplifies to \( \frac{1}{2(x-8)} \). So, the common denominator would be \(2(x-8)(8-x)\). However, \((8-x) = -(x-8)\), therefore, the least common denominator is \(2(x-8)\).

Step 2: Clear the Fractions

Multiply the entire equation by \(2(x-8)\) to eliminate the fractions. This gives:\[ 2(x-8) \left( \frac{1}{2(x-8)} \right) + 2(x-8) \left( \frac{14}{8-x} \right) = \frac{3}{2} \cdot 2(x-8) \]This simplifies to:\[ 1 + 14\cdot(-1) = 3(x-8) \].

Step 3: Simplify the Equation

Simplify the resulting equation:\[ 1 - 14 = 3(x-8) \]\[ -13 = 3(x-8) \].

Step 4: Solve for x

To solve for \(x\), first expand the right side of the equation:\[ -13 = 3x - 24 \]Add 24 to both sides to isolate terms with \(x\):\[ 11 = 3x \]Divide both sides by 3 to solve for \(x\):\[ x = \frac{11}{3} \].

Key Concepts

Common Denominator

Solving equations involving fractions often requires identifying a common denominator. This is crucial because it allows us to combine the fractions into a single equation without denominators, making the equation simpler to solve. A common denominator is the least common multiple of all the denominators in the equation.

• In our example, the denominators are \(2x-16\) and \(8-x\).
• We rewrote \(2x-16\) as \(2(x-8)\) and noted that \(8-x\) is essentially \(-(x-8)\).
• Thus, the least common denominator (LCD) is \(2(x-8)\).

By finding this LCD, we eliminate the fractions in the equation by multiplying every term by \(2(x-8)\). This step makes the equation linear, which simplifies the process of solving it.

Fractions

Fractions are numbers that express ratios between two quantities: a numerator and a denominator. When fractions appear in equations, like in our example, they can make the solving process more complex. Here's what you need to know about handling fractions:

• Fractions like \(\frac{1}{2x-16}\) involve variable denominators, adding complexity to equations.
• Clearing fractions by multiplying through by the common denominator simplifies the problem, turning it into a form without fractions.
• This is why finding a common denominator, as explained, is a critical step into simplifying and solving the equation effectively.

By understanding and managing fractions well, it's easier to transform a complicated equation into one that is straightforward to solve.

Linear Equations

Linear equations are equations that, when graphed, form a straight line. They are characterized by variables raised to the power of one and no higher. Solving linear equations is a key skill:

• Once fractions are eliminated using the common denominator, a linear equation emerges, like the one we simplified to \(-13 = 3(x-8)\).
• Solving linear equations involves isolating the variable. In our case, distributing and combining like terms allowed us to isolate \(x\).
• Finally, simple arithmetic solves for the variable, providing the solution such as \(x = \frac{11}{3}\).

With linear equations, each step has a clear, logical progression, making them easier to solve compared to more complex nonlinear equations. Understanding every part of this process is key to mastery in solving these types of problems.