15. choose efficient methods a test has 50 questions, with 25 questions worth 1 point each and 25 questions worth 3 points each. julia had no more than 20 points subtracted from the total possible points. write an inequality that shows the possible points, p, that julia earned.

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Accepted Answer

# Explanation:
## Step1: Calculate total possible points
First, find the total possible points from the test. There are 25 questions worth 1 point each and 25 questions worth 3 points each. So the total possible points is $25\times1 + 25\times3$.
$25\times1 = 25$, $25\times3 = 75$, so $25 + 75 = 100$.

## Step2: Set up the inequality for Julia's points
Julia had no more than 20 points subtracted from the total possible points. Let $p$ be the points Julia earned. Then the number of points subtracted is $100 - p$, and this is no more than 20. So the inequality is $100 - p\leq20$. We can also rewrite this as $p\geq100 - 20$, which simplifies to $p\geq80$. But we can also express it directly from the subtraction: $p\geq100 - 20$ or starting from the total and subtraction: $100 - p\leq20$ can be rearranged to $p\geq80$, but also, since the minimum she could earn is when she loses 20, so $p\geq100 - 20$, and the maximum is 100 (if she loses 0). But the problem says "no more than 20 points subtracted", so $100 - p\leq20$, which is equivalent to $p\geq80$. However, we can also write the inequality as $p\geq(25\times1 + 25\times3)- 20$, which simplifies to $p\geq100 - 20$, so $p\geq80$. But let's check the initial setup: total points is 100, points subtracted is $100 - p$, and this is $\leq20$, so $100 - p\leq20$, which can be written as $p\geq80$. But also, since she can't earn more than 100, $p\leq100$, but the problem is about the possible points she earned with no more than 20 subtracted, so the main inequality is $p\geq80$ (since $100 - p\leq20$ implies $p\geq80$) and $p\leq100$, but the question is to write an inequality that shows the possible points. So combining, $80\leq p\leq100$, but let's see the steps again.

Wait, let's do it step by step:

1. Calculate total possible points: $25\times1 + 25\times3 = 25 + 75 = 100$.

2. Let $p$ be Julia's score. The number of points subtracted is $100 - p$.

3. The problem states she had no more than 20 points subtracted, so $100 - p\leq20$.

4. Solving for $p$: subtract 100 from both sides: $-p\leq20 - 100$, $-p\leq - 80$, multiply both sides by -1 (remember to reverse the inequality sign): $p\geq80$.

But also, since she can't earn more than the total possible points, $p\leq100$. So the inequality is $80\leq p\leq100$, but the problem might just want the lower bound with the subtraction. Wait, let's check the problem statement: "Write an inequality that shows the possible points, $p$, that Julia earned."

So total possible points: $25(1) + 25(3) = 100$. Points subtracted: $100 - p$. No more than 20 subtracted: $100 - p\leq20$. So the inequality is $p\geq100 - 20$, which is $p\geq80$, and also $p\leq100$ (since she can't earn more than total). But maybe the problem expects the inequality based on the subtraction: $100 - p\leq20$, which can be written as $p\geq80$. Alternatively, $p\geq(25\times1 + 25\times3)-20$, so $p\geq80$.

# Answer:
$p \geq 80$ (or $100 - p \leq 20$ which is equivalent, but $p \geq 80$ is a more direct inequality for her earned points)

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Question

Explanation:

Step1: Calculate total possible points

Step2: Set up the inequality for Julia's points

Answer: