Problem 51 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possible $$\frac{1}{x}+\frac{1}{x+3}=\frac{1}{4}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2\) and \(x = 3\).

Step 1: Simplify the equation

The form of the equation given is a fraction equation. To simplify, the common term for the denominators \(x\) and \(x + 3\) of fractions should be found. The common term for the given fractions is \(x(x + 3)\). So the equation becomes: \[x(x + 3)\left(\frac{1}{x}+\frac{1}{x+3}\right) = \frac{x(x + 3)}{4}\] which simplifies to: \[(x + 3) + x = \frac{x(x + 3)}{4}\]

Step 2: Simplify further and rearrange

Continue by simplifying the equation which leads to: \[2x + 3 = \frac{x^2 + 3x}{4}\]. Now to get rid of the fraction, multiply through by 4 giving: \[8x + 12 = x^2 + 3x\]

Step 3: Rearrange into quadratic form and solve

Rearrange the terms to form a quadratic equation, \(x^2 - 5x + 12 = 0\). Now, factors the equation to solve for \(x\). The factors of the equation are \(x - 3\) and \(x - 2\) which gives \(x = 3\) and \(x = 2\) respectively.

Step 4: Check the solution

Substitute \(x = 2\) and \(x = 3\) into the original equation to validate the answers.

Key Concepts

Fraction Equations

Solving fraction equations involves equations where variables are in the denominators of fractions. Fraction equations can seem tricky at first, but they can be simplified using systematic steps. The goal is to eliminate the fractions, which often requires finding a common denominator.
This means rewriting the fractions in such a way that they all share the same bottom number, or denominator. In our example, the equation \(\frac{1}{x} + \frac{1}{x+3} = \frac{1}{4}\) required eliminating the fractions.
To do so effectively, you need to multiply all parts of the equation by the least common denominator, which simplifies the problem and turns it into a standard equation without fractions. In this case, multiplying every term by \(x(x+3)\) helped us to clear the fractions and allowed us to proceed with traditional solving methods.

Common Denominators

A common denominator is essential when combining fractions, especially in fraction equations. It is the smallest number that can be evenly divided by each of the given denominators. In other words, it's a number each of the denominators will "fit into" exactly.
For the exercise mentioned above, the original fractions \(\frac{1}{x}\) and \(\frac{1}{x+3}\) have different denominators. To find the common denominator, you take the product of both denominators, like \(x(x+3)\).

• This allows for combining the fractions.
• By using \(x(x+3)\), the equation becomes easier to handle.

Once you find a common denominator, you "clear" the fractions by multiplying each term by this denominator, leading to an equation free of fractions, simplified and ready for further transformations.

Quadratic Equations

After eliminating fractions, many fraction equations transform into quadratic forms. A quadratic equation typically looks like \(ax^2 + bx + c = 0\). In our case, the simplified equation was \(8x + 12 = x^2 + 3x\), which then was rearranged to \(x^2 - 5x + 12 = 0\).
Quadratic equations can be solved using several methods, including factoring, completing the square, or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

• For this exercise, factoring was used, which involves rewriting the quadratic as two binomial expressions that equal zero.
• This allows you to solve for each solution separately.

Have confidence when tackling quadratic equations; they are a significant and standard part of algebra and offer many methods to find solutions.

Simplification of Irrational Solutions

Irrational solutions, such as those involving square roots, can occur when solving equations, especially quadratics. These solutions are numbers that cannot be expressed as a simple fraction. In this exercise, though, the solutions were rational (\(x = 2\) and \(x = 3\)), there is still potential for irrational solutions in other similar problems.
Simplifying irrational solutions involves transforming them into their simplest form. This might mean representing a square root in its simplest components or expressing radicals in the most reduced form possible. Below is how you can approach this:

• Check if the number under the square root in the solution is a perfect square.
• Break down the number into prime factors if needed, to simplify.

While this problem provided rational solutions, developing skills to simplify irrational solutions is crucial, as they are common in various advanced math topics.