Problem 37 - 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 \(39-46\).) $$\frac{x}{3}+\frac{1}{2}=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -3\).

Step 1: Identify the Denominators

Examine the equation \(\frac{x}{3}+\frac{1}{2}=-\frac{1}{2}\) and note the denominators: 3 and 2. These fractions have different denominators that need to be addressed.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest number that both denominators can divide into without a remainder. For 3 and 2, the LCD is 6.

Step 3: Multiply Entire Equation by LCD

Multiply every term in the equation \(\frac{x}{3}+\frac{1}{2}=-\frac{1}{2}\) by the LCD, which is 6, to eliminate the fractions.\[6 \times \left( \frac{x}{3} \right) + 6 \times \left( \frac{1}{2} \right) = 6 \times \left( -\frac{1}{2} \right)\]

Step 4: Simplify Each Term

Perform the multiplication.\[2x + 3 = -3\]Here, \(6 \div 3 = 2\) and \(6 \div 2 = 3\); apply these to each fraction.

Step 5: Solve for \(x\)

Now, solve the equation for \(x\):Subtract 3 from both sides:\[2x = -3 - 3\]This simplifies to:\[2x = -6\]Divide both sides by 2:\[x = -3\]

Key Concepts

Solving Equations

When dealing with equations, especially those that include fractions, our goal is to find the value of the variable that makes the entire equation true. Solving equations can seem complex, but breaking down each step helps clarify the process. The first thing to do is to analyze the equation and ensure every step follows logically.

Consider the exercise we're working on: \( \frac{x}{3} + \frac{1}{2} = -\frac{1}{2} \). Our variable, \(x\), is part of terms that include fractions. We need to simplify this situation to isolate \(x\). The overall idea is to eliminate the fractions by using a common denominator, then perform basic arithmetic operations like addition, subtraction, multiplication, or division to solve the equation.

• Identify and remove constraints: These are the fractions, and we remove them by multiplying by their least common denominator (LCD).
• Simplify the operations: Once the fractions are gone, it's simpler to isolate the variable and solve for it as we do in simpler algebraic equations.

Least Common Denominator

The least common denominator is a key concept when solving equations with fractions. It is the smallest number that can be evenly divided by all the denominators in an equation. This common denominator allows us to clear fractions from the equation, making it significantly easier to solve.

To determine the least common denominator, look at the denominators in the equation. In our example \( \frac{x}{3} + \frac{1}{2} = -\frac{1}{2} \), the denominators are 3 and 2. We need the smallest number that both 3 and 2 divide into without leaving a remainder. A useful strategy is to list the multiples of each denominator:

• Multiples of 3: 3, 6, 9, 12, ...
• Multiples of 2: 2, 4, 6, 8, ...

From these, the smallest common multiple is 6, making it the LCD. Using the LCD, we multiply each term of the equation to eliminate the fractions. This simplification transforms the equation into a format where you can easily solve for the variable.

Fractions in Equations

Fractions often make equations look intimidating, but if you know how to handle them, they become less daunting. When an equation includes fractions, the main aim is to eliminate these fractions for an easier calculation. At first, it involves combining operations with the least common denominator to clear them out of the way.

For instance, let's apply what we learned to the example \( \frac{x}{3} + \frac{1}{2} = -\frac{1}{2} \). By multiplying each fraction by the least common denominator (in this case, 6), you're leveling the playing field:

• Multiply \( \frac{x}{3} \) by 6 to get \(2x\).
• Multiply \( \frac{1}{2} \) by 6 to get 3.
• Multiply \( -\frac{1}{2} \) by 6 to get -3.

After clearing the fractions, the equation becomes \( 2x + 3 = -3 \). These steps transform the equation into a simpler, no-fraction form. From here, proceed with basic algebra: subtract 3 from each side to yield \(2x = -6\), then divide both sides by 2, leading to \(x = -3\). The visualization of working through these simplifications reinforces a solid understanding of managing fractions in equations.