Problem 31 - Algebra A Combined Function

Question

Solve each equation. $$ \frac{2}{x-2}+1=\frac{x}{x+2} $$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Identify the least common denominator (LCD)

To eliminate the denominators from the equation, we first identify the least common denominator (LCD). The denominators in the equation are \(x-2\) and \(x+2\). Their LCD is \((x-2)(x+2)\).

Step 2: Multiply the entire equation by the LCD

Next, multiply each term in the equation by the LCD, \((x-2)(x+2)\), to clear the fractions. This gives us:\[ (x-2)(x+2) \left( \frac{2}{x-2} \right) + (x-2)(x+2)(1) = (x-2)(x+2) \left( \frac{x}{x+2} \right) \]

Step 3: Simplify the equation

Simplify each term:1. \( (x+2) imes 2 = 2(x+2) = 2x + 4 \)2. \( (x-2)(x+2) \times 1 = x^2 - 4 \)3. \( (x-2) imes x = x(x-2) = x^2 - 2x \)So, the equation becomes:\[ 2x + 4 + x^2 - 4 = x^2 - 2x \]

Step 4: Combine like terms

Combine like terms on both sides of the equation. Start by simplifying both sides:\( 2x + x^2 - 4 + 4 = x^2 - 2x \)The equation reduces to:\( 2x + x^2 = x^2 - 2x \)

Step 5: Isolate variable terms

Subtract \(x^2\) from both sides to eliminate \(x^2\):\( 2x = -2x \)Add \(2x\) to both sides to isolate \(x\):\( 4x = 0 \)

Step 6: Solve for x

Divide both sides by 4 to find \(x\):\[ x = 0 \]

Step 7: Verify the solution

Substitute \(x = 0\) back into the original equation to verify:\[ \frac{2}{0-2} + 1 = \frac{0}{0+2} \]The left side becomes -1 + 1 = 0, and the right side is 0.Since both sides equal, \(x = 0\) is indeed the solution.

Key Concepts

Least Common Denominator

When working with rational equations, it's important to eliminate fractions to simplify the problem. One way to achieve this is by finding the least common denominator (LCD).
The LCD is essentially the smallest expression that can be evenly divided by each of the denominators present in the equation. In our example, the denominators are \(x-2\) and \(x+2\).
By multiplying each term in the equation by the LCD,

• we convert the entire equation into one without fractions,
• making it easier to manage.

In essence, the LCD ensures each term of the rational equation is treated equally, laying a solid groundwork for further simplification and solving. In this case, the LCD \((x-2)(x+2)\) sets the stage for eliminating all fractional components.

Solving Rational Equations

Rational equations can often appear complex at first glance due to the presence of fractions. Thankfully, there is a structured approach to solving them.
Starting with the least common denominator (LCD), we multiply each term by it to simplify the equation, as seen in our example. By doing this:

• we remove fractions,
• leaving us with polynomial equations, which are easier to handle.

Once fractions are cleared, follow these steps: 1. Expand each term to unpack the equation further.
2. Rearrange the equation so that all similar terms are grouped together, preparing us for combining like terms.
Treat each step as a building block, where simplifying the equation through clear rules allows us to identify and isolate the variable. Ultimately, this leads to finding the value of the variable that satisfies the equation.

Combine Like Terms

In algebra, combining like terms is a key principle when solving equations. Once a rational equation is free from fractions and properly expanded, you'll notice terms that can be simplified by combining like terms.
To "combine like terms," follow these guidelines:

• Identify terms that have the same variable with the same exponent. These are your like terms.
• Add or subtract them as necessary to simplify the equation.

In our exercise, we have terms like \(2x\) and \(x^2\), which we handle separately since they contain different variables. After simplification, you're left with a clearer path to solve the equation.
This process not only simplifies equations but also helps in isolating variables, making it easier to solve for unknowns. Combining like terms is a fundamental skill, crucial for simplifying expressions and equations efficiently.