To find the correct representations of the inequality \( 6x \geq 3 + 4(2x - 1) \), we should first simplify the inequality.
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Start with the original inequality: \[ 6x \geq 3 + 4(2x - 1) \]
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Distribute \( 4 \) inside the parentheses: \[ 6x \geq 3 + 8x - 4 \]
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Combine the constants on the right side: \[ 6x \geq 8x - 1 \]
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Rearranging the inequality (subtract \( 8x \) from both sides): \[ 6x - 8x \geq -1 \] This simplifies to: \[ -2x \geq -1 \]
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Dividing both sides by -2 (remember to flip the inequality sign): \[ x \leq \frac{1}{2} \]
Now, let's evaluate the options for correctness:
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Option 1: \( 1 \geq 2x \)
- This can be rearranged to \( x \leq \frac{1}{2} \). This option is correct.
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Option 2: \( 6x \geq 3 + 8x - 4 \)
- This is the same form we derived when we simplified the original inequality. This option is correct.
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Option 3: A number line from -1.5 to 1.5, a point at 0.5, and a bold line pointing to the left.
- This indicates \( x \leq 0.5 \), which is a correct representation.
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Option 4: A number line from -1.5 to 1.5, a point at -0.5, and a bold line pointing to the right.
- This indicates \( x \geq -0.5 \), which is incorrect based on our derived inequality.
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Option 5: A number line from -1.5 to 1.5, a point at 0.5, and a bold line pointing to the right.
- This indicates \( x \geq 0.5 \), which is also incorrect based on our derived inequality.
Based on the analysis, the three correct representations are:
- \( 1 \geq 2x \)
- \( 6x \geq 3 + 8x - 4 \)
- The number line with a point at 0.5 and a bold line pointing to the left (indicating \( x \leq 0.5 \)).
5 answers