Problem 39 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{x+1}{3 x+9}+\frac{x}{2 x+6}=\frac{2}{4 x+12}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 0.2\).

Step 1: Simplify the Equation

Notice that each of the denominators, \(3x + 9\), \(2x + 6\), and \(4x + 12\), are multiples of \(x + 3\). So they can be simplified: \[\frac{x+1}{3(x + 3)}+\frac{x}{2(x + 3)}=\frac{2}{4(x + 3)}\]which can be further simplified to:\[\frac{x+1}{3}+\frac{x}{2}=\frac{2}{4}\]

Step 2: Find the least common denominator (LCD)

The denominators are now 3, 2, and 4. The least common denominator of these is 12.

Step 3: Multiply each term by the LCD

Do this to get rid of the fractions:\[(x+1) * 4 + x * 6 = 2 * 3\]

Step 4: Solve the resulting equation

We now have a simple linear equation, which we can solve for \(x\):\[4x + 4 + 6x = 6\]Combine like terms:\[10x + 4 = 6\]Subtract 4 from both sides:\[10x = 2\]Finally, divide both sides by 10 to solve for \(x\):\[x = \frac{2}{10} = 0.2\]

Key Concepts

Understanding the Least Common Denominator

When dealing with rational equations, finding the least common denominator (LCD) is a crucial step. The LCD is the smallest number that all denominators can divide into without leaving a remainder. This concept is similar to finding the least common multiple but focuses solely on denominators.

To find the LCD in our problem, we start by identifying the current denominators of the simplified equation: 3, 2, and 4. The goal is to determine the smallest number that each of these can divide by evenly.

Here's a quick tip to find the LCD:

• List the multiples of each denominator.
• Look for the smallest common multiple among them.

In this case, we find that 12 is the least common multiple of 3, 2, and 4, making it the LCD. Once identified, you can use the LCD to eliminate the fractions and work towards solving the equation.

Simplification of Fractions

Simplifying fractions is an important precursor to solving equations efficiently. This involves reducing fractions to their simplest form, by finding a common factor in the numerator and the denominator and dividing each by this number.

In the given exercise, the denominators were initially expressions: \(3x + 9\), \(2x + 6\), and \(4x + 12\). Each can be factored further:

• \(3x + 9 = 3(x + 3)\)
• \(2x + 6 = 2(x + 3)\)
• \(4x + 12 = 4(x + 3)\)

This simplification reveals a common factor \((x + 3)\) in each denominator. By canceling \((x + 3)\) from all fractions, we simplify the equation to: \[\frac{x+1}{3} + \frac{x}{2} = \frac{2}{4}\]This simplification makes it easier to find the LCD and solve the equation.

Solving Linear Equations

Once the rational equation is free of fractions, the path to solving it becomes clearer. You're left with an equation that's linear, meaning it takes the form \(ax + b = c\), where the solution is a single value for \(x\).

After clearing the fractions by multiplying each term by the LCD (in our case, 12), you get:\[(x+1) \times 4 + x \times 6 = 2 \times 3\]This simplifies further to a linear equation:\[4x + 4 + 6x = 6\]Combine like terms (\(4x + 6x\)) to get \(10x\) plus 4:\[10x + 4 = 6\]
Now, isolate \(x\) by moving the constant term (4) to the other side:\[10x = 2\]Then, divide by 10 to solve for \(x\):\[x = \frac{2}{10} = 0.2\]With practice, solving linear equations becomes straightforward, leading to solutions that satisfy the original rational equation.