To find an equivalent expression to \( 5(3x + 9) \), we'll start by simplifying it:
\[ 5(3x + 9) = 5 \cdot 3x + 5 \cdot 9 = 15x + 45 \]
Next, let's see if any of the given options can be manipulated to yield that expression.
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a) \( 5x + 65 = 240 \)
- Solve for \( x \): \[ 5x = 240 - 65 \] \[ 5x = 175 \quad \Rightarrow \quad x = 35 \]
- This does not match \( 15x + 45 \).
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b) \( 5x - 65 = 240 \)
- Solve for \( x \): \[ 5x = 240 + 65 = 305 \quad \Rightarrow \quad x = 61 \]
- This does not match \( 15x + 45 \).
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c) \( 65 - 5x = 240 \)
- Solve for \( x \): \[ -5x = 240 - 65 \quad \Rightarrow \quad -5x = 175 \quad \Rightarrow \quad x = -35 \]
- This does not match \( 15x + 45 \).
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d) \( 5x + 240 = 65 \)
- Solve for \( x \): \[ 5x = 65 - 240 \quad \Rightarrow \quad 5x = -175 \quad \Rightarrow \quad x = -35 \]
- This does not match \( 15x + 45 \).
None of the options a), b), c), or d) provide an equivalent expression to \( 15x + 45 \).
Thus, we need to find a connection between these expressions and the simplified form \( 15x + 45 \).
None of the equations in the current options are equivalent to \( 5(3x + 9) \). Therefore, there is no option that is equivalent to \( 5(3x + 9) \) from the choices given.
If we consider \( 15x + 45 \) as a function of \( x \), we can see each option produces a specific value of \( x \), but does not yield \( 15x + 45 \) inherently.
In conclusion, the mathematically equivalent form of \( 5(3x + 9) \) is \( 15x + 45 \), and none of the provided options represent this equivalence.