Question

The equation $$ \frac{5}{x+3}+3=\frac{8+x}{x+3} $$ has no solution, yet when we go through the process of solving it, we obtain \(x=-3 .\) Write a brief paragraph to explain what causes this to happen.

Short answer

The equation has no solution because \(x = -3\) results in a denominator of zero, which is undefined.

Step-by-step solution

Identify the Denominator

Observe that the common denominator in both fractions is \(x+3\).

Simplify Both Sides

Multiply both sides of the equation by \(x+3\) to eliminate the denominator: \[5 + 3(x+3) = 8 + x.\]

Expand and Simplify

Expand and simplify the equation: \[5 + 3x + 9 = 8 + x.\] Combine like terms: \[3x + 14 = 8 + x.\]

Isolate x

Solve for \(x\) by subtracting \(x\) and \(8\) from both sides: \[3x - x + 14 - 8 = 0\]. Simplify: \[2x + 6 = 0\].

Solve for x

Divide both sides by \2\ to find \(x\): \[x = -3\].

Check for Validity

Substitute \(x = -3\) back into the original equation. This leads to a denominator of 0 in the fractions, which is undefined. Thus, \(x = -3\) is not a valid solution.

Conclusion

The equation has no solution because \(x = -3\) causes division by zero, making it invalid in the context of the original fractions.

Key concepts

Undefined Expressions

When dealing with rational equations, it's crucial to recognize conditions that make the equation undefined. In this exercise, the denominator is \(x+3\). If \(x\) equals \(-3\), the denominator becomes zero, causing the fractions to be undefined. Division by zero is undefined in mathematics, which means any value that causes a denominator to become zero is not a valid solution. Hence, as soon as we determine that \(x = -3\), we must conclude it makes the fractions undefined. Therefore, double-checking for these conditions is essential before finalizing a solution.

Common Denominators

When solving rational equations, finding a common denominator simplifies the process. Here, both denominators are already the same: \(x + 3\). This allows us to eliminate them easily. To simplify, we multiply every term by the common denominator, \(x + 3\). This step removes the fractions and results in: \[5 + 3(x+3) = 8 + x.\] After eliminating the fractions, the problem becomes a straightforward algebraic equation. Always ensure the denominators are the same before eliminating them.

Checking Solutions

After finding a potential solution to a rational equation, it's vital to verify it. Substitute the value back into the original equation to see if it holds true. For our problem, substituting \(-3\) results in a zero denominator, making the expression undefined: \(\frac{5}{-3+3} +3 = \frac{8-3}{-3+3}\). Since division by zero is not allowed, \(x=-3\) is not a solution. Therefore, checking our work helps us avoid invalid solutions. Always substitute potential solutions back into the original equation to confirm their validity.