Problem 38 - 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 \(39-46\).) $$\frac{x}{2}+\frac{4}{3}=-\frac{2}{3}$$

Step-by-Step Solution

Answer

The solution is \(x = -4\).

Step 1: Identify all denominators

The given equation is \( \frac{x}{2} + \frac{4}{3} = -\frac{2}{3} \). The denominators in this equation are \(2\) and \(3\).

Step 2: Find the Least Common Denominator (LCD)

The least common denominator of \(2\) and \(3\) is \(6\) because it is the smallest number that both \(2\) and \(3\) can divide evenly.

Step 3: Multiply entire equation by the LCD

Multiply each term in the equation by the LCD, \(6\): \[6 \times \frac{x}{2} + 6 \times \frac{4}{3} = 6 \times \left(-\frac{2}{3}\right)\].

Step 4: Simplify the Equation

Perform the multiplication to simplify:1. \(6 \times \frac{x}{2} = 3x\)2. \(6 \times \frac{4}{3} = 8\)3. \(6 \times \left(-\frac{2}{3}\right) = -4\)Thus, the equation simplifies to: \(3x + 8 = -4\).

Step 5: Solve for x

Subtract \(8\) from both sides to isolate the term with \(x\):\[3x = -4 - 8\]\[3x = -12\].

Step 6: Divide by the Coefficient of x

Divide both sides by \(3\) to solve for \(x\):\[x = \frac{-12}{3}\]\[x = -4\].

Key Concepts

Least Common Denominator (LCD)

When tackling equations that include fractions, especially when solving fractional equations, it's essential to first identify the least common denominator (LCD). Think of it as the smallest number that can be evenly divided by all the denominators present in the problem. This step simplifies the entire process, allowing you to eliminate the fractions and transform the equation into a more manageable form.
For instance, in the equation \( \frac{x}{2} + \frac{4}{3} = -\frac{2}{3} \), we have two denominators: \(2\) and \(3\). The LCD here is \(6\), as it is the smallest number that both \(2\) and \(3\) divide evenly into. By using the LCD, you simplify the equation, turning it from a fractional equation into a linear one without fractions, which is usually much easier to solve.

Solving Linear Equations

Once the fractions are removed by multiplying the entire equation by the LCD, you're left with a linear equation. Solving linear equations involves finding the value of the variable that makes the equation true. This typically involves the following steps:

• Combine like terms if necessary to simplify the equation further.
• Isolate the variable by getting all terms involving the variable on one side and all constant terms on the other.
• Perform inverse operations to solve for the variable, such as addition/subtraction and multiplication/division.

After multiplying the original equation by \(6\), we solve \(3x + 8 = -4\) by moving 8 to the other side of the equation. This gives us \(3x = -12\). Dividing both sides by the coefficient of \(x\), which is \(3\), reveals that \(x = -4\).
Understanding these steps is crucial as it forms the foundation for solving more complex equations.

Fractional Equations

Handling fractional equations requires a good grasp of fractions and basic algebra. A fractional equation is any equation that has fractions in it, which can make straightforward operations like addition or subtraction a bit tricky.
To solve fractional equations effectively, always aim to eliminate the fractions first. This is where finding the LCD comes into play. By multiplying every term by the LCD, you clear out the fractions, transforming the equation into a form that's easier to work with. Take, for example, \(6 \times \frac{x}{2} = 3x\); here, multiplying by 6 cancels the denominator 2.

Mastering the art of solving fractional equations will not only help you in algebra but also in higher-level math where more complex fractional relationships are explored. It's all about recognizing the form and applying the right techniques to simplify the equation.