Problem 41 - Algebra 2

Question

Write two rational expressions whose sum is \(\frac{x-2}{x+4}\)

Step-by-Step Solution

Answer

The two rational expressions are \( \frac{1}{x+4} \) and \( \frac{x-3}{x+4} \).

Step 1: Initial Assumption

Let's start by assuming the two rational expressions as \( \frac{a}{x+4} \) + \( \frac{b}{x+4} \). Here, \(a\) and \(b\) are constants that we need to identify.

Step 2: Addition of Rational Expressions

By adding the two expressions together, we get \( \frac{a+b}{x+4} \). Now, this should be equal to the given expression \( \frac{x-2}{x+4} \).

Step 3: Equating the Numerator

From step 2, we see that \(a+b\) must be equal to \(x-2\) as the denominators are already equal. This implies that a + b = x - 2.

Step 4: Determining the Values of a and b

We know that a and b are constants and x is a variable. So for a + b to be equal to x - 2, we can assign any suitable values to a and b as long as they satisfy the equation. Let's choose a = 1 and b = x - 3. This makes a + b = 1 + (x - 3) = x - 2, which satisfies the equation.

Step 5: Providing the Solution

So, the two rational expressions whose sum is \( \frac{x-2}{x+4} \)are \( \frac{1}{x+4} \) and \( \frac{x-3}{x+4} \).

Key Concepts

Addition of Rational Expressions

When dealing with rational expressions, understanding how to add them is crucial. Don't be intimidated by the complex look of these terms. Let's break it down. Rational expressions are similar to fractions, involving numerators and denominators.
If two rational expressions have the same denominator, you can directly add their numerators. For instance, adding \( \frac{a}{x+4} \) and \( \frac{b}{x+4} \) keeps the denominator \( x+4 \) the same. The addition happens in the numerators, giving us \( \frac{a+b}{x+4} \).

• Make sure the denominators are the same before combining.
• Just like regular fractions, you add or subtract the numerators.
• The common denominator stays the same.

This concept becomes much simpler once you know you’re only manipulating the numerators, provided the denominators are identical.

Solving Equations

Solving equations often involves finding unknown values that satisfy the given conditions. In the context of rational expressions, we're typically trying to determine values of constants that match a specific condition, as seen in the original problem.
To illustrate, let's consider how the problem asks you to find values of \( a \) and \( b \) for the equation \( a+b=x-2 \). Solving this involves choosing values for the constants that satisfy this condition. Notice how cleverly assigning values helps you achieve the desired outcome.

• Identify the equation you need to solve.
• Assign values that satisfy all parts of the equation.
• Check your assignments by substituting back into the original context.

This approach is a detective-like exercise in algebra, balancing the equation by carefully selecting suitable values.

Algebraic Expressions

Algebraic expressions form the backbone of many mathematical problems. They include variables, constants, and operations like addition and subtraction. Simplifying these expressions often requires combining like terms or identifying patterns.In the context of rational expressions, keep an eye on the structure of both numerators and denominators. Here, identifying that \( a + b = x - 2 \) allows you to equate terms wisely, simplifying the expression while respecting its structure.

• Understand that variables can take any permissible value, while constants remain fixed.
• Look for patterns that help simplify or rearrange the expressions.
• Use logical reasoning to decide how best to approach an equation or expression.

Remember, whether you’re simplifying, adding, or solving, breaking down complex algebraic expressions helps in recognizing the familiar patterns and solutions.