Questions: Solve the equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 3(x+2)=2+3x 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 B. The solution set is x x is a real number . C. The solution set is ∅.

Solve the equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 3(x+2)=2+3x 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 B. The solution set is x x is a real number . C. The solution set is ∅.
Transcript text: Solve the equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. \[ 3(x+2)=2+3 x \] 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 $\square$ B. The solution set is $\{x \mid x$ is a real number $\}$. C. The solution set is $\varnothing$.
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Solution

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Solution Steps

Step 1: Expand the left side of the equation

Expand \( 3(x + 2) \) using the distributive property: \[ 3(x + 2) = 3x + 6 \] The equation becomes: \[ 3x + 6 = 2 + 3x \]

Step 2: Subtract \( 3x \) from both sides

Subtract \( 3x \) from both sides to simplify: \[ 3x + 6 - 3x = 2 + 3x - 3x \] This simplifies to: \[ 6 = 2 \]

Step 3: Analyze the result

The equation \( 6 = 2 \) is false, which means there is no solution to the original equation. Therefore, the equation is inconsistent.

Step 4: Determine the type of equation

Since the equation has no solution, it is an inconsistent equation.

Step 5: Select the correct choice

The correct choice is: \[ \text{C. The solution set is } \varnothing. \]

Final Answer

The correct answer is C. The solution set is \( \varnothing \).

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