Problem 57 - Elementary Algebra

Question

For the following problems, solve the rational equations. $$ \frac{2}{x}=\frac{3}{x+2}+1 $$

Step-by-Step Solution

Answer

Answer: x ≈ -1.388

Step 1: Combine Fractions

We can rewrite the equation to have a common denominator for all terms: $$ \frac{2}{x} = \frac{3}{x+2} + \frac{x+2}{x+2} $$ Now, we can combine the fractions on the right-hand side: $$ \frac{2}{x} = \frac{3 + x(x+2)}{x+2} $$

Step 2: Find the Least Common Denominator (LCD)

In this case, the LCD is the product of the denominators x and (x+2): $$ LCD = x(x+2) $$

Step 3: Multiply Both Sides by the LCD

Multiplying both sides by the LCD to eliminate fractions: $$ x(x+2) \cdot \frac{2}{x} = x(x+2) \cdot \frac{3 + x(x+2)}{x+2} $$

Step 4: Simplify the Equation

Now, we can simplify the equation by canceling out terms on both sides: $$ 2(x+2) = x(3 + x(x+2)) $$ Expanding and simplifying the resulting equation: $$ 2x+4 = 3x + x^3 + 2x^2 $$ Rearrange the equation to have all the terms on one side: $$ x^3 + 2x^2 -x - 4 = 0 $$

Step 5: Solve for x

Unfortunately, this equation does not have an obvious factorization or an easy way to solve algebraically. However, we can use numerical methods (such as trial and error, a graphing calculator, or specialized software) to obtain an approximate value of x. Using a graphing calculator or software, we find that the approximate solution for x is: $$ x \approx -1.388 $$

Key Concepts

Common Denominator

Rational equations can be tricky due to the fractions involved. When you see fractions in an equation, the first step is typically to rewrite them with a common denominator. This simplifies the equation by putting everything on the same playing field. Think of it as leveling the ground for comparison. For the given equation

• \( \frac{2}{x} = \frac{3}{x+2} + 1 \)

we start by making the right-hand side have a common denominator. This involves turning the whole number 1 into a fraction with a denominator of \( x+2 \). Hence, 1 becomes \( \frac{x+2}{x+2} \), allowing us to add it to \( \frac{3}{x+2} \). Now, both fractions can neatly fit together under a single denominator, making them easier to handle. This concept of a common denominator is fundamental when dealing with rational equations, as it allows all terms to be unified for further manipulation.

Least Common Denominator (LCD)

Once you have your common denominators sorted out, it's time to find the Least Common Denominator (LCD) for the entire equation. The LCD is crucial because it helps eliminate the fractions entirely when you multiply through by it. In essence, multiplying both sides of the equation by the LCD gets rid of those pesky denominators so you can focus on solving the polynomial equation that remains.

In the equation

• \( \frac{2}{x} = \frac{3 + x(x+2)}{x+2} \)

the LCD is found by multiplying together all unique denominators, \( x \) and \( x+2 \), giving us \( x(x+2) \). This step is key because it converts our rational equation into a polynomial one that is usually much simpler to solve, at least in form. The act of multiplying through by the LCD cleans the slate, allowing the problem to be tackled from a familiar algebraic angle.

Solving Polynomial Equations

After applying the LCD to clear the fractions, you're left with a pure polynomial equation. This step converts the problem into familiar territory. Polynomial equations come in many forms, and solving them requires different strategies.

For the simplified equation

• \( 2x+4 = 3x + x^3 + 2x^2 \)

we combine like terms to bring everything to one side, resulting in \( x^3 + 2x^2 -x - 4 = 0 \). This resembles the general form \( ax^n + bx^{n-1} + \ldots + c = 0 \).

This specific equation doesn't factor easily or have obvious roots. When faced with such a polynomial, numerical methods or graphing approaches are employed to find approximate solutions. Techniques such as using graphing calculators or software provide solutions like \( x \approx -1.388 \). Understanding these methods enhances problem-solving skills by expanding the toolbox beyond basic algebraic manipulation.