Problem 16 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ \frac{2}{y+2}-\frac{y}{2-y}=\frac{y^{2}+4}{y^{2}-4} $$

Step-by-Step Solution

Answer

Any \( y \neq -2, 2 \) satisfies the equation since these ensure a valid common denominator.

Step 1: Identify the Problem

We are given the equation \( \frac{2}{y+2} - \frac{y}{2-y} = \frac{y^2+4}{y^2-4} \) to solve. Our task is to find values of \( y \) that satisfy this equation.

Step 2: Recognize the Denominators

Observe that the denominators \( y+2 \), \( 2-y \) and \( y^2-4 \) suggest restrictions on \( y \). Since \( y^2-4 = (y+2)(y-2) \), each possible denominator must not be zero. Hence, \( y eq -2 \) or \( y eq 2 \).

Step 3: Identify Common Denominator

The common denominator combining all terms is \((y+2)(2-y)\), which is \(-(y^2-4)\). Recognizing that \( y^2-4 = (y+2)(y-2) \), the common denominator can also be written as \(-1 \cdot (y^2-4)\).

Step 4: Adjust Equivalent Fractions

Rewrite each fraction over the common denominator. Left hand side: \( \frac{2(2-y) + y(y+2)}{-(y^2-4)} \). Right hand side stays \( \frac{y^2+4}{y^2-4} \).

Step 5: Simplify the Equation

Combine the numerators: \( 2(2-y) = 4 - 2y \) and \( y(y+2) = y^2 + 2y \). Thus, the left side becomes \( - (y^2 + 2y - 2y + 4) = -( y^2 - 4) \) on \(-(y^2-4) = y^2-4\).

Step 6: Set Numerators Equal

The revised equation is \(-1(y^2 - 4) = (y^2 + 4)\), indicating equivalence because both sides convert to \(-1(y^2-4)\).

Step 7: Analyze Resulting Equation

Since we've found \(-(y^2-4) = -(y^2-4)\), this means any \( y eq -2, 2 \) will satisfy the equation.

Step 8: Verify Solution Set

By exclusion of \( y = -2, 2 \), confirm every other \( y \) resolves the equation \((y=-1,0,1,...etc.\)), as they abide by the original setup's restrictions.

Key Concepts

Common Denominator

Finding a common denominator is an essential technique when working with fractions in algebra. It allows you to combine fractions by expressing them with the same denominator. In the original exercise, the fractions have denominators related to the equation's variables. The denominators are \( y+2 \), \( 2-y \), and \( y^2-4 \). To find the common denominator, you should first factor anything you can. Here, recognize that \( y^2-4 = (y+2)(y-2) \). This makes sense because it allows you to combine the terms into a single expression.

To manage these expressions better, you note that \( 2-y \) can be seen as \(-1(y-2)\). As you aim for a common denominator, rewrite \((y+2) \) and \((2-y) \) in terms of something related to \((y^2-4)\), which results in \(-(y^2-4)\) after recognizing it as the same set of factors. With this common ground, you're set to convert initial fractions for further equation handling.

Equivalent Fractions

Equivalent fractions are fractions that have different numerators and denominators but represent the same value. In solving rational equations, converting expressions into equivalent fractions is key to solving them.

In the equation presented, moving both sides to use the common denominator \(-(y^2-4)\) allows the fractions to be comparable. This means adjusting the numerators appropriately. For instance, the term \( \frac{2}{y+2} \) becomes \( \frac{2(2-y)}{-(y^2-4)} \) as it adopts the new denominator. This transformation maintains equivalence as you don't change the original value, merely its representation.

Once each fraction is expressed with this common denominator, you're equipped to simplify and solve the equation because you're comparing like terms. Remember, focusing on writing equivalent fractions can simplify the complexity of many algebraic problems.

Rational Equations

Rational equations are equations that include fractions with variables in their denominators. Solving these equations involves finding the values of the variable that make the equation true while ensuring no denominators are zero. The exercise introduces you to a basic rational equation where the variable \( y \) resides in multiple denominators.

Begin by understanding restrictions due to the denominators. Solving \( y^2-4 \) as a base for denominators leads to restrictions: \( y eq 2 \) and \( y eq -2 \). These critical restrictions arise because a zero denominator makes the term undefined.

After setting up with a common denominator, rational equations can often come down to simpler algebraic expressions. The given equation simplifies via comparing numerators due to having adjustable equivalent fractions. By making the left-hand and right-hand side numerators equal under consolidated terms, you find solutions that fulfill these rational equations effectively while adhering to restrictions.

Simplifying Expressions

Simplifying expressions is a fundamental part of solving rational equations. It requires both combining like terms and reducing equations to their simplest form, often making the problem easier to solve.

In this exercise, after applying a common denominator across terms, combine terms and simplify on each side. For instance, you recognize on the left that \( 2(2-y) \) forms \( 4 - 2y \), while \( y(y+2) \) simplifies to \( y^2 + 2y \). Together these form a simplified numerator, leading to \( -(y^2 - 4) \) after adjustments.

This type of simplification turns a complex equation into something more manageable. When simplified fully, the equality \(-(y^2-4) = (y^2+4)\) yields meaningful comparisons by turning complicated algebraic expressions into clearer numerical or algebraic insights through simplification.