To find out which expression can be written in the box such that the equation has no solution, we first simplify the left side of the original equation:
\[ 3 (x + 1) + 9 \]
Distributing the 3 gives:
\[ 3x + 3 + 9 \]
Combining like terms:
\[ 3x + 12 \]
Now the equation looks like this:
\[ 3x + 12 = \text{box} \]
For the equation to have no solution, the expression on the right side (the box) must be a constant that never equals \(3x + 12\), meaning it cannot equal the linear expression \(3x + 12\) for any value of \(x\). This happens when the box is a different constant that does not intersect with any value of the left side.
Let’s evaluate each of the given options to see if they could lead to no solutions:
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3 (x + 4): \[ 3(x + 4) = 3x + 12 \] This would lead to a solution, so not applicable.
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2 (x + 6) + x: \[ 2(x + 6) + x = 2x + 12 + x = 3x + 12 \] This again leads to a solution, so not applicable.
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4(x – 3) – x: \[ 4(x - 3) - x = 4x - 12 - x = 3x - 12 \] This simplifies to \(3x - 12\). Setting this equal to \(3x + 12\), we get: \[ 3x - 12 = 3x + 12 \Rightarrow -12 = 12 \] This is a contradiction and represents no solution.
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3 (x + 1) + 9 x: \[ 3(x + 1) + 9x = 3x + 3 + 9x = 12x + 3 \] This leads to a linear function not related to the original one, but it will have solutions for various \(x\).
So, the expression that can be written in the box such that the equation has no solution is:
4(x – 3) – x.