Problem 34 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{2}{x+1}+\frac{3}{x-2} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{5x - 1}{(x+1)(x-2)}\), with \(x \neq -1, 2\).

Step 1: Find the Common Denominator

Here, the denominators are \(x+1\) and \(x-2\). So, the common denominator is the product of these two, i.e., \((x+1)(x-2)\).

Step 2: Rewrite the Fractions

Now, rewrite each fraction with the common denominator and simplify the numerators:\n\[\frac{2}{x+1} \times \frac{x-2}{x-2} + \frac{3}{x-2} \times \frac{x+1}{x+1} = \frac{2(x-2)}{(x+1)(x-2)} + \frac{3(x+1)}{(x+1)(x-2)} \]

Step 3: Simplify the Fractions

Combine the fractions now that they have the same denominator: \[ \frac{2(x-2) + 3(x+1)}{(x+1)(x-2)} \] Now, simplify the numerator: \[ \frac{2x - 4 + 3x + 3}{(x+1)(x-2)} = \frac{5x - 1}{(x+1)(x-2)} \]

Step 4: Check the Restrictions

Also, the simplified form must also keep the same restrictions as the original ones. Here, since the expression is undefined when \(x = -1\) or \(x = 2\), these two values should be excluded from the solution set.

Key Concepts

Common Denominator

Adding algebraic fractions can be tricky, especially when they have different denominators. To make them compatible for addition, a common denominator is necessary. Think of a common denominator in algebra like a shared term for fractions, allowing you to combine them seamlessly.

In our example, the denominators are \(x+1\) and \(x-2\). The best approach to find a common denominator is to multiply these two distinct terms together, resulting in \((x+1)(x-2)\). This new denominator works because it includes all factors from both original denominators, enabling the fractions to be expressed with this single, unified base.

This step is crucial for simplifying expressions and allows you to proceed with combining the numerators easily. Always remember to adjust each fraction so that they can share this common ground.

Simplifying Expressions

Once a common denominator has been found, the next step is simplifying the expression. This involves rewriting each fraction with the new common denominator, which means adjusting the numerators accordingly.

For example, to adjust \(\frac{2}{x+1}\), we multiply both the numerator and the denominator by \(x-2\), resulting in \(\frac{2(x-2)}{(x+1)(x-2)}\). Likewise, for \(\frac{3}{x-2}\), multiply by \(x+1\), giving \(\frac{3(x+1)}{(x+1)(x-2)}\).

With both fractions now sharing the common denominator, you can combine them:

• Add the adjusted numerators: \(2(x-2) + 3(x+1)\).
• Distribute and simplify: \(2x - 4 + 3x + 3 = 5x - 1\).

Finally, the expression simplifies to \(\frac{5x - 1}{(x+1)(x-2)}\). Simplification is like tidying up—ensuring everything is neatly expressed.

Rational Expressions

Rational expressions are ratios of polynomials, similar to fractions in arithmetic. They play a pivotal role in algebra, modeling various real-world scenarios and mathematical concepts.

When dealing with rational expressions, it's important to understand not just how to manipulate them but also their constraints. In our example, although the simplified expression \(\frac{5x - 1}{(x+1)(x-2)}\) looks compact, it inherits restrictions from the original fractions.

The expression is undefined where the original denominators would lead to division by zero, specifically at \(x = -1\) or \(x = 2\). These are values where the product \((x+1)(x-2)\) equals zero, causing the entire fraction to be invalid.

• Whenever you simplify a rational expression, check for these values.
• These restrictions are crucial in math problems that involve real-world applications where values might be physically or logically impossible.

Understanding this ensures that when you work with rational expressions, you avoid inaccuracies and logical errors.