Question

Determine whether the statement is true or false. If the statement is​ false, make the necessary​ change(s) to produce a true statement.
The equation 2 left parenthesis x plus 4 right parenthesis equals 2 left parenthesis 4 plus x right parenthesis has precisely one solution.
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Part 1
Choose the correct answer below.
A.
The statement is false. The equation 2 left parenthesis x plus 4 right parenthesis equals 2 left parenthesis 4 plus x right parenthesis is true for all values of x.
B.
The statement is true.
C.
The statement is false. The equation 2 left parenthesis x plus 4 right parenthesis equals 2 left parenthesis 4 plus x right parenthesis has no solution.
D.
The statement is false. The equation 2 left parenthesis x plus 4 right parenthesis equals 2 left parenthesis 4 plus x right parenthesis has precisely one solution.

Answer 1

Let's analyze the equation given in the statement:

\[
2(x + 4) = 2(4 + x)
\]

First, we can simplify both sides of the equation:

The left side:
\[
2(x + 4) = 2x + 8
\]

The right side:
\[
2(4 + x) = 2 \cdot 4 + 2 \cdot x = 8 + 2x
\]

Now we rewrite the equation using the simplified expressions:
\[
2x + 8 = 8 + 2x
\]

Next, we can simplify further by subtracting \(2x\) from both sides:
\[
8 = 8
\]

This is a true statement that holds for all values of \(x\). Thus, the equation does not have a unique solution; instead, it is an identity that is true for every \(x\).

Therefore, the statement "The equation has precisely one solution" is **false**. The accurate conclusion is that the equation is true for all values of \(x\).

Thus, the correct choice is:
**A. The statement is false. The equation 2(x + 4) = 2(4 + x) is true for all values of x.**