Question

Solve and check the following linear equation.
19equals4​(xminus4​)minus​(xminus8​)
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Part 1
What is the​ solution? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is StartSet nothing EndSet
  
.
B.
The solution set is StartSet x font size decreased by 3 | font size decreased by 3 x is a real number EndSet.
C.
The solution set is empty set.

Answer 1

To solve the equation \( 19 = 4(x - 4) - (x - 8) \), let's first simplify the right-hand side.

1. Distribute the \(4\) in \(4(x - 4)\):
\[
4(x - 4) = 4x - 16
\]

2. Distribute the \(-1\) in \(-(x - 8)\):
\[
-(x - 8) = -x + 8
\]

3. Combine these results:
\[
19 = (4x - 16) + (-x + 8)
\]
Simplifying further:
\[
19 = 4x - 16 - x + 8
\]
\[
19 = 4x - x - 16 + 8
\]
\[
19 = 3x - 8
\]

4. Now, isolate \(x\) by adding \(8\) to both sides:
\[
19 + 8 = 3x
\]
\[
27 = 3x
\]

5. Divide by \(3\) to solve for \(x\):
\[
x = \frac{27}{3} = 9
\]

So, the solution is \(x = 9\).

### Check:
Let's substitute \(x = 9\) back into the original equation to verify:

Left-hand side:
\[
19
\]

Right-hand side:
\[
4(9 - 4) - (9 - 8)
\]
Calculating:
\[
4(5) - (1) = 20 - 1 = 19
\]

Both sides are equal, confirming the solution is correct.

### Conclusion:
Since the solution \(x = 9\) is valid and is a single solution, we can conclude that the correct representation of the solution set is:

**Option B: The solution set is \(\{ 9 | x \text{ is a real number} \}\).**