Questions: Solve the equation for (x). [ 3(x-2)-3=5 x-2(4+x) ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (x=) □ (Type an integer or a simplified fraction.) B. The solution is all real numbers. C. There is no solution.

Solve the equation for (x). [ 3(x-2)-3=5 x-2(4+x) ] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (x=) □ (Type an integer or a simplified fraction.) B. The solution is all real numbers. C. There is no solution.
Transcript text: Solve the equation for $x$. \[ 3(x-2)-3=5 x-2(4+x) \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $x=$ $\square$ (Type an integer or a simplified fraction.) B. The solution is all real numbers. C. There is no solution.
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Solution

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Solution Steps

To solve the equation \(3(x-2)-3=5x-2(4+x)\), we will first expand both sides to eliminate the parentheses. Then, we will collect like terms and isolate \(x\) on one side of the equation to solve for it. Finally, we will check if the solution is valid or if there are any special cases such as no solution or all real numbers being solutions.

Step 1: Expand Both Sides

We start with the equation: \[ 3(x-2) - 3 = 5x - 2(4+x) \] Expanding both sides gives: \[ 3x - 6 - 3 = 5x - 8 - 2x \] This simplifies to: \[ 3x - 9 = 3x - 8 \]

Step 2: Rearrange the Equation

Next, we rearrange the equation to isolate terms involving \(x\): \[ 3x - 9 - 3x = -8 \] This simplifies to: \[ -9 = -8 \]

Step 3: Analyze the Result

The equation \(-9 = -8\) is a contradiction, indicating that there are no values of \(x\) that satisfy the original equation. Therefore, the solution set is empty.

Final Answer

The answer is C. There is no solution.

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