To determine what additional operation should be included in the expression \(54x + 6\) to make it equivalent to \(12x + 6\), we can set up the equation:
\[ 54x + 6 + \text{(additional operation)} = 12x + 6 \]
Subtracting \(6\) from both sides gives us:
\[ 54x + \text{(additional operation)} = 12x \]
Now we can isolate the additional operation:
\[ \text{(additional operation)} = 12x - 54x \] \[ \text{(additional operation)} = -42x \]
Now we need to assess the provided options and see which one will yield an equivalent expression.
The options are:
- \( 34x \)
- \( \frac{3}{4} x \)
- \( -34x \)
- \( -\frac{3}{4} x \)
- \( -x \)
- \( -2x \)
None of those options add up to \(-42x\) directly, but let's check the option that contributes the closest reduction:
- If we take \( -34x\), then:
\[ 54x + 6 - 34x + 6 = 20x + 6 \quad (\text{not equivalent}) \]
- If we take \( -\frac{3}{4} x\):
\[ 54x + 6 - \frac{3}{4} x = 53.25x + 6 \quad (\text{not equivalent}) \]
- If we take \( -x\):
\[ 54x + 6 - x = 53x + 6 \quad (\text{not equivalent}) \]
- If we take \( -2x \):
\[ 54x + 6 - 2x = 52x + 6 \quad (\text{not equivalent}) \]
None of the provided options yield \( -42x\) to simplify to \(12x\).
It seems the correct operation that should be performed to achieve \( 12x + 6\) is not among the options given based on provided options.
Thus, the original intention may require revisiting the options. If we need to approximate to the available options and want the greatest decrement, the closest might be using \( -34x \) even though not exactly producing \( 12x + 6\).
In a direct challenge to match exactly \( -42x \), it seems none suffice.
The best "adjustment" that might contribute towards simplifying close ideas would generally lead us at:
However, the answer to the question from the provided options isn't exact; no additional value is applicable based on this exact requirement with the choices. Though collectively must note \( -34x\) may be mathematically closest; reporting on the specified aim seeks payback each operational must concisely set this delivery upon thus you may want direct subtraction represented correctly along those lines.
If you are constrained strictly to select from the available responses, choose:
- -34x; although keep in mind this is an adjustment toward the closest understanding or remaining evaluative standards furthemore wouldn't yield \( 12x + 6\) minimum exact point valuing.
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