Problem 17 - College Algebra

Question

Solve each rational equation for x. State all x-values that are excluded from the solution set. \(2-\frac{3}{x+4}=\frac{x+2}{x+4}\)

Step-by-Step Solution

Answer

The solution is \(x = -3\) with \(x = -4\) being excluded.

Step 1: Identify Excluded Values

To determine which values of \(x\) are excluded from the solution set, we need to identify when the denominator of any fraction becomes zero. The fractions in this equation have the denominator \(x + 4\). So, solve \(x + 4 = 0\) which gives \(x = -4\) as the excluded value.

Step 2: Clear Fractions from the Equation

Multiply every term by \(x + 4\) to eliminate the denominators. This results in the equation:\(\ (2 - \frac{3}{x+4}) * (x+4) = (\frac{x+2}{x+4} * (x+4)\)Which simplifies to:\(2(x + 4) - 3 = x + 2\).

Step 3: Distribute and Simplify

Distribute the \(2\) on the left side of the equation:\(2x + 8 - 3 = x + 2\).Simplify the equation:\(2x + 5 = x + 2\).

Step 4: Solve for x

Rearrange the equation to isolate \(x\) on one side:Subtract \(x\) from both sides: \(2x - x + 5 = x - x + 2\) which simplifies to \(x + 5 = 2\).Subtract \(5\) from both sides: \(x + 5 - 5 = 2 - 5\) which simplifies to \(x = -3\).

Key Concepts

Excluded Values

When solving rational equations, it's crucial to determine the values that could potentially make the equation undefined. These are called "excluded values." For any fraction, the denominator cannot be zero because division by zero is undefined. In this particular equation, we have fractions with the denominator \(x + 4\). This means we need to determine when \(x + 4\) equals zero. By solving the simple equation \(x + 4 = 0\), we find that \(x = -4\) is an excluded value. Therefore, \(x = -4\) is a value for which the original equation does not work, as it would lead to division by zero.It is important to always begin by identifying these values before proceeding with any further steps in solving the equation, as they help us understand the boundaries of our solution set. After finding these values, make a note of them as they will determine which solutions are valid.

Clearing Fractions

Fraction elimination is a key step when dealing with rational equations. We call this process "clearing fractions." In rational equations, we often multiply each term by the common denominator to eliminate the fractions altogether.In the equation given, both sides have the denominator \(x + 4\). By multiplying every term in the equation by \(x + 4\), we effectively cancel out the denominators:

• On the left-hand side, \( (2 - \frac{3}{x+4})\cdot(x+4) \), the fraction disappears, simplifying it to \(2(x+4) - 3\).
• On the right-hand side, \(\frac{x+2}{x+4}\cdot(x+4)\), simplifies it directly to \(x+2\).

This step transforms a rational equation into an algebraic equation, making it easier to solve by traditional means.

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to simplify expressions. It states that \(a(b + c) = ab + ac\). In this problem, after clearing the fractions, we have \(2(x + 4) - 3 = x + 2\).Using the distributive property here:

• Distribute the \(2\) across \(x + 4\), resulting in \(2 \cdot x + 2 \cdot 4\).
• Simplify to get \(2x + 8\).
• The equation now looks like \(2x + 8 - 3 = x + 2\).

After applying the distributive property, the equation is much simpler, setting us up beautifully for further steps of isolating the variable to solve for \(x\). Remembering the distributive process is hugely beneficial in breaking down complex algebraic expressions simplistically.

Isolation of Variables

Isolating the variable is the decisive step in solving equations and involves rearranging the equation so that the unknown variable appears by itself on one side.In our case, after simplifying through distribution, you have:\(2x + 5 = x + 2\).To isolate \(x\):

• Subtract \(x\) from both sides to get \(2x - x + 5 = x - x + 2\), simplifying to \(x + 5 = 2\).
• Next, subtract \(5\) from both sides: \(x + 5 - 5 = 2 - 5\).
• This process gives us \(x = -3\).

The variable \(x\) is now isolated on the left, giving its value. It's important to check our found solutions against any excluded values determined earlier—in this case, \(x = -3\) isn't excluded, so it stands as a valid solution.