To determine which expressions are equivalent to the original algebraic expression \( (3x - 8) + (2x + 5) - (4x - 8) \), let's simplify the expression step-by-step.
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Start with the given expression: \[ (3x - 8) + (2x + 5) - (4x - 8) \]
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Distribute the negative sign in the third term: \[ (3x - 8) + (2x + 5) - 4x + 8 \]
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Now combine like terms:
- Combine the \( x \) terms: \[ 3x + 2x - 4x = (3 + 2 - 4)x = 1x = x \]
- Combine the constant terms: \[ -8 + 5 + 8 = (-8 + 8) + 5 = 0 + 5 = 5 \]
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Putting it all together, we simplify the entire expression: \[ x + 5 \]
Now, let's check which of the given responses are equivalent to \( x + 5 \):
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Response: \( 3x - 8 + 2x + 5 - 4x - 8 \)
- This simplifies to \( (3 + 2 - 4)x + (-8 + 5 - 8) = 1x + (-11) = x - 11 \) (not equivalent)
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Response: \( 3x - 8 + 2x + 5 - 4x + 8 \)
- This simplifies to \( (3 + 2 - 4)x + (-8 + 5 + 8) = 1x + 5 = x + 5 \) (equivalent)
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Response: \( 3x + 2x - 4x - 8 + 5 + 8 \)
- This simplifies to \( (3 + 2 - 4)x + (-8 + 5 + 8) = 1x + 5 = x + 5 \) (equivalent)
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Response: \( 5x - 4x - 8 + 5 + 8 \)
- This simplifies to \( (5 - 4)x + (-8 + 5 + 8) = 1x + 5 = x + 5 \) (equivalent)
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Response: \( x - 8 + 5 + 8 \)
- This simplifies to \( x + (-8 + 5 + 8) = x + 5 \) (equivalent)
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Response: \( x - 13 + 8 \)
- This simplifies to \( x + (-13 + 8) = x - 5 \) (not equivalent)
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Response: \( x - 3 + 8 \)
- This simplifies to \( x + (-3 + 8) = x + 5 \) (equivalent)
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Response: \( x - 21 \)
- This does not match \( x + 5 \) (not equivalent)
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Response: \( x + 5 \)
- This is obviously equivalent (equivalent)
Conclusion
The equivalent expressions from the given options are:
- \( 3x - 8 + 2x + 5 - 4x + 8 \)
- \( 3x + 2x - 4x - 8 + 5 + 8 \)
- \( 5x - 4x - 8 + 5 + 8 \)
- \( x - 8 + 5 + 8 \)
- \( x - 3 + 8 \)
- \( x + 5 \)
So, the equivalent expressions are:
- \( 3x - 8 + 2x + 5 - 4x + 8 \)
- \( 3x + 2x - 4x - 8 + 5 + 8 \)
- \( 5x - 4x - 8 + 5 + 8 \)
- \( x - 8 + 5 + 8 \)
- \( x - 3 + 8 \)
- \( x + 5 \)