Problem 75 - Elementary Algebra

Question

Find the sum. \(\frac{x-7}{x+5}+\frac{x+4}{x-2}\).

Step-by-Step Solution

Answer

Answer: \(\frac{2x^2+34}{(x+5)(x-2)}\).

Step 1: Identify the denominators

In this exercise, we have two fractions: 1. \(\frac{x-7}{x+5}\), with denominator \(x+5\). 2. \(\frac{x+4}{x-2}\), with denominator \(x-2\).

Step 2: Find the common denominator

To find the common denominator, we need to multiply the two denominators together, which are \((x+5)\) and \((x-2)\). Therefore, the common denominator is \((x+5)(x-2)\).

Step 3: Adjust both fractions for the common denominator

Now, we need to adjust both fractions so that the denominators are equal to the common denominator. 1. For the first fraction \(\frac{x-7}{x+5}\), we need to multiply the numerator and denominator by \((x-2)\). This will give us: \(\frac{(x-7)(x-2)}{(x+5)(x-2)}\). 2. For the second fraction \(\frac{x+4}{x-2}\), we need to multiply the numerator and denominator by \((x+5)\). This will give us: \(\frac{(x+4)(x+5)}{(x-2)(x+5)}\).

Step 4: Add the adjusted fractions

Now that we have fractions with the same denominator, we can add them together: \(\frac{(x-7)(x-2)}{(x+5)(x-2)}+\frac{(x+4)(x+5)}{(x-2)(x+5)}\). To find the sum, add the numerators and keep the common denominator: \(\frac{(x-7)(x-2)+(x+4)(x+5)}{(x+5)(x-2)}\).

Step 5: Simplify the numerator

Expand the expressions in the numerator: \(\frac{(x^2-9x+14)+(x^2+9x+20)}{(x+5)(x-2)}\). Now, combine the like terms in the numerator: \(\frac{2x^2+34}{(x+5)(x-2)}\). We cannot simplify any further, so the final answer is: \(\frac{2x^2+34}{(x+5)(x-2)}\).

Key Concepts

Common Denominator

When working with fractions, it's essential to have the same denominator if you want to add or subtract them. The term 'common denominator' refers to the denominator that all fractions in an expression share. To find it, especially in algebraic fractions like in this example, multiply the unique factors from each fraction's denominators.

• For our example, the denominators are \(x+5\) and \(x-2\).
• To find a common denominator, we multiply these two terms: \((x+5)(x-2)\).

By using the common denominator, we can rewrite each fraction so that they can be directly compared and combined. This process is a crucial skill as fractions with different denominators cannot be added or subtracted directly without first finding this shared base.

Rational Expressions

Rational expressions are similar to rational numbers, but they involve polynomials in their numerators and/or denominators. Just like fractions, rational expressions represent the division of two expressions. They require treatment similar to numeric fractions when operations such as addition, subtraction, multiplication, and division are involved.

• In our exercise, we deal with \(\frac{x-7}{x+5}\) and \(\frac{x+4}{x-2}\) which are both rational expressions.
• These expressions consist of polynomials like \(x-7\), \(x+5\), etc., in their numerators and denominators.

Handling rational expressions involves familiar processes such as finding common denominators, simplifying expressions, and ensuring expressions are in their simplest form. Remember, simplifying is key to making it easier to work with these expressions.

Fraction Addition

Adding fractions, whether they're numeric or algebraic like our rational expressions, follows a straightforward process once you have a common denominator. After finding this denominator, adjust each fraction accordingly.

• In our example, this involved multiplying to equalize the denominator across the expressions.
• Once adjusted, as in our step-by-step solution, you combine the numerators over the common denominator.

The final step is straightforward - you simply add the adjusted numerators, keeping the common denominator the same.
Properly simplifying the sum helps to ensure the expression is easily understandable and usable in further calculations or solutions.