Question

Solve the equation. Then determine whether the equation is an​ identity, a conditional​ equation, or an inconsistent equation.
4xplus27equals9​(xplus3​)minus5x
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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 equation has a single solution. 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.
Part 2
What type of equation is​ this?
A.
an identity
B.
a conditional equation
C.
an inconsistent equation

Answer 1

To solve the equation \(4x + 27 = 9(x + 3) - 5x\), we will first simplify both sides.

Starting with the right side:

\[
9(x + 3) - 5x = 9x + 27 - 5x = 4x + 27
\]

Now we can set the left side equal to the right side:

\[
4x + 27 = 4x + 27
\]

Notice that both sides of the equation are identical. This indicates that the equation is true for any value of \(x\).

### Part 1: Solution
Since the equation \(4x + 27 = 4x + 27\) is true for all \(x\), it suggests that the solution set contains all real numbers.

The appropriate choice is:
**B. The solution set is StartSet x | x is a real number EndSet.**

### Part 2: Type of Equation
Because the equation holds true for all \(x\), it is considered an identity.

The appropriate choice is:
**A. an identity.**

### Summary
- The solution set is the set of all real numbers.
- The equation is an identity.