Problem 17 - College Algebra with Corequisite Support

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 \( x = -4 \) excluded.

Step 1: Identify the denominator

First, observe the denominators of the rational expressions in the equation. The denominators are both \(x+4\). Identify these to determine the values of \(x\) that are excluded from the solution set.

Step 2: State excluded values

The denominator cannot be zero. Therefore, set \(x+4=0\) and solve for \(x\). This gives us \(x=-4\). So, \(x=-4\) is excluded from the solution set.

Step 3: Clear the denominators

Multiply every term of the equation by \(x+4\) to eliminate the denominators:\[(x+4)\left(2 - \frac{3}{x+4}\right) = (x+4)\left(\frac{x+2}{x+4}\right)\].

Step 4: Simplify the equation

Distribute and simplify:\[2(x+4) - 3 = x + 2\].This simplifies to:\[2x + 8 - 3 = x + 2\].Which further simplifies to:\[2x + 5 = x + 2\].

Step 5: Solve for x

Subtract \(x\) from both sides to isolate terms with \(x\):\[2x - x + 5 = x - x + 2\].This results in:\[x + 5 = 2\].Finally, subtract 5 from both sides:\[x = 2 - 5\], resulting in:\[x = -3\].

Key Concepts

Solving Rational Equations

When dealing with rational equations, our main goal is to find the value of the variable that satisfies the equation. Rational equations involve fractions where the numerator and/or denominator contains a polynomial. Here’s how you can solve these types of equations effectively:

Start by identifying the denominators present in the equation. This is crucial, as the denominator is what makes the equation "rational." For instance, in the equation \(2 - \frac{3}{x+4} = \frac{x+2}{x+4}\), the common denominator is \(x+4\). Recognizing this helps in both solving the equation and finding excluded values.

Once you've identified the denominators, you can eliminate them by multiplying every term in the equation by the least common multiple of the denominators. In our example, multiplying through by \(x+4\) gives us \((x+4)(2 - \frac{3}{x+4}) = (x+4)(\frac{x+2}{x+4})\). This clears the equation of fractions, simplifying it to a linear equation, which is then easier to solve.

Finally, simplify and solve for the variable like a regular linear equation. For our particular problem, this involved expanding and simplifying terms to eventually isolate the variable, yielding \(x = -3\). By removing the fractions early on, you make the solution straightforward and manageable.

Excluded Values in Solutions

In rational equations, excluded values play a critical role because they represent the values of the variable that would make the denominator zero. Remember, division by zero is undefined, and thus these values cannot be part of the solution.

The first step is to identify the denominators in the equation. For our problem, the denominator \(x+4\) is present. To determine the excluded values, set the denominator equal to zero and solve for the variable. This calculation gives us \(x+4 = 0\), and solving this equation yields \(x = -4\).

This value is excluded from the solution set. In simpler terms, if \(x\) were \(-4\), the rational expressions involved would be undefined. Therefore, it’s essential to perform this check before solving the equation, ensuring that any solution found does not include these problematic values.

Algebraic Denominators

Algebraic denominators are part of what defines a rational equation. These are expressions that contain a variable, like \(x+4\), which appears in the denominator of a fraction. Understanding and working with them is crucial for solving rational equations successfully.

First, note that with algebraic denominators, variables can affect the domain of the expression because they create the potential for zero in the denominator. This is why recognizing and excluding certain values is vital, as those that make the denominator zero must be explicitly stated as exclusions.

By identifying the algebraic denominator, you can determine the excluded values, which informs both the process of solving the equation and verifying the solution set. It also assists in clearing the denominators from the equation by multiplying each term by this denominator, making it easier to handle and solve the equation. This methodology reduces a complex rational equation to a simpler form, typically leading to a straightforward solution.