To solve the inequality \(2x - 1 \geq 11\), we can first isolate \(x\):
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Add 1 to both sides: \[ 2x \geq 12 \]
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Divide both sides by 2: \[ x \geq 6 \]
This means that John needs to buy an item that costs at least 6 dollars (or he could buy 2 items for at least 12 dollars total if there's a promotion).
Now, let's analyze the situations provided:
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John has a budget of 11 dollars. A promotion is going on where he can get a dollar off if he buys 2 of the same item for x dollars.
- For this situation, the total cost with the promotion (if he buys 2 items) would be \(2x - 1\). We would check: Is it possible for \(2x - 1\) to be greater than or equal to 11? Solving \(2x - 1 \geq 11\) leads to \(x \geq 6\), which could be valid, but he can only spend 11 dollars total.
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John needs to spend more than 11 dollars. A promotion is going on where he can get 2 dollars off if he buys 2 of the same item for x dollars.
- In this case, we check if \(2x - 2 \geq 11\) (because of the 2 dollars off). Rearranging gives \(2x \geq 13\), or \(x \geq 6.5\). This indicates John needs to spend more than 11 dollars under the promotion, which fits well with \(2x - 1 \geq 11\).
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John has a budget of 2 dollars. A promotion is going on where he can get a dollar off if he buys 11 of the same item for x dollars.
- Here, if John buys 11 items, he would effectively pay \(11x - 1\) and might not satisfy the context of needing at least $11 (due to the budget of $2).
Considering the inequalities derived, the second situation is the only one that matches the originally given inequality \(2x - 1 \geq 11\):
John needs to spend more than 11 dollars. A promotion is going on where he can get 2 dollars off if he buys 2 of the same item for x dollars.