Problem 45 - Prealgebra

Question

Solve each equation by first finding the LCD for the fractions in the equation and then multiplying both sides of the equation by it.(Assume \(x\) is not 0 in Problems \(39-46\).) $$\frac{1}{x}-\frac{1}{2}=-\frac{1}{4}$$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 4 \).

Step 1: Identify the Fractions

We have the equation \( \frac{1}{x} - \frac{1}{2} = -\frac{1}{4} \). The fractions present are \( \frac{1}{x} \), \( \frac{1}{2} \), and \( -\frac{1}{4} \).

Step 2: Determine the LCD

The denominators in the equation are \( x \), \( 2 \), and \( 4 \). To find the least common denominator (LCD), we take the least common multiple of these numbers, which is \( 4x \).

Step 3: Multiply the Equation by the LCD

Multiply every term in the equation by \( 4x \) to eliminate the fractions: \[ 4x \left( \frac{1}{x} \right) - 4x \left( \frac{1}{2} \right) = 4x \left( -\frac{1}{4} \right) \]This simplifies to:\[ 4 - 2x = -x \]

Step 4: Simplify the Equation

Simplify the obtained equation from Step 3:\[ 4 - 2x = -x \] Add \(-x\) to both sides to get:\[ 4 - 2x + x = 0 \] Which simplifies to:\[ 4 - x = 0 \]

Step 5: Solve for x

Solve the simplified equation for \( x \):Add \( x \) to both sides:\[ 4 = x \] Thus, \( x = 4 \).

Key Concepts

Least Common Denominator (LCD)

Understanding the Least Common Denominator (LCD) is crucial when solving rational equations like \(\frac{1}{x} - \frac{1}{2} = -\frac{1}{4}\). When dealing with fractions, the denominators (the numbers or expressions under the fraction line) can make equations challenging to solve. The LCD is the smallest number that all the denominators can divide without leaving a remainder. This number helps you remove the fractions from the equation by turning them into whole numbers.
To find the LCD, look at all the fractions involved and list out their denominators. For this problem, those denominators are \(x\), \(2\), and \(4\). The key is to look for the smallest number that is a multiple of each denominator. In this case, the LCD is \(4x\) because \(4x\) is the smallest expression that each of the denominators divides evenly. Multiplying the equation by the LCD allows each fraction to simplify, leading to easier calculation with whole numbers.

Multiplying Fractions

Multiplying both sides of an equation by the LCD is a strategy used to eliminate fractions. Fractions complicate equations, so eliminating them makes solving the equation much simpler. Here's how you do it.
Once the LCD is identified, as with our example where it is \(4x\), multiply every single term in the equation by \(4x\). This technique uses the property of "clearing fractions."
For instance, let's multiply each term in \(\frac{1}{x} - \frac{1}{2} = -\frac{1}{4}\) by \(4x\):

• \(4x \left( \frac{1}{x} \right) = 4\)
• \(-4x \left( \frac{1}{2} \right) = -2x\)
• \(4x \left( \frac{-1}{4} \right) = -x\)

This transforms the equation into \(4 - 2x = -x\), removing all fractions and making it easier to solve.

Simplifying Equations

Once fractions are cleared, the next step is simplifying the equation. Simplification involves using basic algebraic principles to make the equation as straightforward as possible. You do this by combining like terms and moving variables to one side of the equation while moving constants to the other side.
With our example reduced to \(4 - 2x = -x\), we combine like terms. To move \(-x\) from the right side, add \(x\) to both sides, resulting in \(4 - 2x + x = 0\). This simplifies further to \(4 - x = 0\).
The goal is to isolate \(x\) on one side. By adding \(x\) to each side, you obtain \(4 = x\). Thus, the solution is \(x = 4\), meaning that when you substitute \(4\) back into the original equation, each part balances, confirming the solution is correct.