Problem 17 - College Algebra

Question

For the following exercises, 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 excluded value \(x=-4\).

Step 1: Identify the common denominator

The given rational equation is \(2 - \frac{3}{x+4} = \frac{x+2}{x+4}\). Notice that both fractions share a common denominator of \(x+4\). This is important because it will help us eliminate the fractions in the next step.

Step 2: Clear the fractions

Multiply each term of the equation by the common denominator \(x+4\) to eliminate the fractions. This gives:\[ (2)(x+4) - 3 = x + 2 \] Simplifying the left side, we have: \[ 2x + 8 - 3 = x + 2 \] which further simplifies to: \[ 2x + 5 = x + 2 \]

Step 3: Solve for \(x\)

Subtract \(x\) from both sides to isolate \(x\) on one side: \[ 2x - x + 5 = x - x + 2 \] Simplifying gives: \[ x + 5 = 2 \] Now, subtract 5 from both sides to solve for \(x\): \[ x = 2 - 5 \] which simplifies to: \[ x = -3 \]

Step 4: Check for excluded values

The only potential excluded value comes from the restriction that the denominator cannot be zero. Thus, set \(x+4 = 0\) and solve for \(x\): \[ x = -4 \] So, \(x = -4\) is excluded from the solution set.

Step 5: Verify the solution

Substitute \(x = -3\) back into the original equation to verify the solution: \[ 2 - \frac{3}{-3+4} = \frac{-3+2}{-3+4} \] Simplifying both sides gives: \[ 2 - 3 = -1 \] and \[ 1 = 1 \] Both sides are equal, confirming that \(x = -3\) is indeed a valid solution.

Key Concepts

Common Denominator

In solving rational equations, identifying a common denominator is crucial. It simplifies the process by allowing us to eliminate fractions in the equation. The common denominator is essentially the lowest common multiple of all denominator expressions in the equation. If two or more fractions share the same denominator, such as \(\frac{3}{x+4}\) and \(\frac{x+2}{x+4}\) in the given problem, that denominator is the common denominator.

When we multiply every term in the equation by this common denominator, we eliminate the fractions. This transforms the equation into a simpler form without fractions. For instance, multiplying every term in our example by \(x+4\) leads to \(2(x+4) - 3 = x+2\), thereby removing the denominators entirely.

This step is beneficial because it reduces the complexity of the equation, making it easier to manipulate and solve in subsequent steps. Remember to apply the multiplication to each term to maintain the equation's balance.

Excluded Values

When solving rational equations, there's an important consideration: excluded values. These are specific values of \(x\) that would make any denominator equal to zero. A zero denominator is undefined in mathematics. Identifying these values is crucial to ensure our solutions are valid.

In our exercise, both fractions have a denominator of \(x+4\). Hence, we must find where this expression equals zero by setting \(x+4=0\) and solving for \(x\). Solving this equation, we find that \(x=-4\).

This means \(-4\) is excluded from our solution set because plugging it into the original equation would result in undefined fractions.

• Always check each denominator in your equation.
• Make sure to compute where these denominators turn zero.
• Exclude these values from your solution set.

Identifying these excluded values not only helps in verifying solutions but also ensures the mathematical integrity of the solution.

Solving Equations

Solving rational equations involves isolating the variable step by step, once the fractions have been cleared. After using the common denominator to eliminate fractions, you simplify and rearrange the equation to solve for the unknown variable.

In our exercise, we simplified \(2(x+4) - 3 = x + 2\) into \(2x + 5 = x + 2\). The next logical step is to isolate \(x\). This involves moving all terms involving \(x\) to one side and constant terms to the other. We subtracted \(x\) from both sides to get \(x + 5 = 2\).

Next, subtract 5 from each side to isolate \(x\), resulting in \(x = -3\).

• Start by eliminating fractions using the common denominator.
• Simplify the equation by combining like terms.
• Isolate \(x\) by performing inverse operations.

Finally, verify your solution by substituting it back into the original equation to ensure both sides are equal. This confirms that you've found a correct and valid solution.