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We are given the following problem:

- Bernadette has 600 quarts of an alcohol solution containing 15% pure alcohol.
- She needs to add *x* quarts of pure alcohol so that the resulting solution has at most 30% pure alcohol.
- We need to find the inequality that represents this situation.

### Breaking down the problem:

1. **Initial amount of alcohol in the solution**:
   - The total amount of alcohol in 600 quarts of a 15% solution is:
     \[
     0.15 \times 600 = 90 \text{ quarts of alcohol}.
     \]

2. **Amount of alcohol added**:
   - Adding *x* quarts of pure alcohol means she is adding exactly *x* quarts of alcohol.

3. **Total volume after adding the pure alcohol**:
   - After adding *x* quarts of pure alcohol, the total volume of the solution becomes \(600 + x\) quarts.

4. **Condition on the final concentration**:
   - We need the final concentration of alcohol to be at most 30%. This leads to the following inequality for the concentration of alcohol:
     \[
     \frac{90 + x}{600 + x} \leq 0.30.
     \]

### Solving the inequality:

Multiply both sides by \(600 + x\) to eliminate the denominator:
\[
90 + x \leq 0.30(600 + x).
\]

Expanding the right-hand side:
\[
90 + x \leq 180 + 0.30x.
\]

Now, subtract \(0.30x\) from both sides:
\[
90 + 0.70x \leq 180.
\]

Subtract 90 from both sides:
\[
0.70x \leq 90.
\]

Finally, divide by 0.70:
\[
x \leq \frac{90}{0.70} = 128.57.
\]

Thus, Bernadette can add at most approximately 128.57 quarts of pure alcohol to keep the solution under 30% alcohol.

### Correct answer:
This inequality corresponds to **option D**: 
\[
90 + x \leq 420 + 0.70x.
\]

Would you like further clarification or details?

Here are five related questions you might find interesting:
1. How would the inequality change if the target concentration was 25% instead of 30%?
2. What if Bernadette started with a different volume, say 500 quarts, but still wanted to reach a 30% concentration?
3. How would the solution change if she added a different liquid containing a certain percentage of alcohol, instead of pure alcohol?
4. What if the concentration limit was set to exactly 30%, instead of "at most" 30%?
5. How can we model the situation if instead of adding alcohol, Bernadette wanted to dilute the solution with water?

**Tip**: When solving concentration problems, always start by determining the total amount of pure substance (in this case, alcohol) before and after the addition, and compare this to the desired concentration.

Bernadette currently has a 600 quart alcohol solution containing 15% pure alcohol. She needs to add at least x quarts of pure alcohol so that the solution has at most 30% solution. Which of the following inequalities best represent the situation?

Learn how to solve an inequality problem involving the addition of pure alcohol to a solution to achieve a desired concentration. This example involves algebraic concepts and percentage calculations, suitable for students in Grades 9-11.

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