Question
Solve the equation: \( \frac{x-4}{x+1}=\frac{8}{x+1} \) \( \{0\} \) \( \{1,12\} \) \( \{-1,12\} \) \( \{12\} \)
Ask by Bryant Bolton. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution is \( x = 12 \).
Solution
Let's solve the equation step by step:
\[
\frac{x - 4}{x + 1} = \frac{8}{x + 1}
\]
**Step 1: Identify Restrictions**
First, ensure that the denominator is not zero:
\[
x + 1 \neq 0 \implies x \neq -1
\]
**Step 2: Simplify the Equation**
Since both sides of the equation have the same denominator and \( x \neq -1 \), you can multiply both sides by \( x + 1 \) to eliminate the denominator:
\[
x - 4 = 8
\]
**Step 3: Solve for \( x \)**
\[
x - 4 = 8 \implies x = 8 + 4 \implies x = 12
\]
**Step 4: Verify the Solution**
Ensure that \( x = 12 \) does not make the denominator zero:
\[
x + 1 = 12 + 1 = 13 \neq 0
\]
So, \( x = 12 \) is a valid solution.
**Conclusion:**
The solution to the equation is:
\[
\{12\}
\]
**Answer:** \(\{12\}\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To solve the equation \( \frac{x-4}{x+1}=\frac{8}{x+1} \), we first note that \( x \neq -1 \) to avoid division by zero. Next, we can multiply both sides by \( x + 1 \) (since \( x + 1 \) is not zero) to eliminate the fractions: \[ x - 4 = 8 \] Now, we solve for \( x \): \[ x = 8 + 4 = 12 \] The only solution we find is \( x = 12 \). Now, let’s consider that since \( x + 1 \) can never be zero, we check for any other potential extraneous solutions. Since \( x = -1 \) would make the original equation undefined, we only consider \( x = 12 \). Thus, the only valid solution is \( \{12\} \).
