Why did you say to subtract on step 2 but you added in the math

You're correct to point out the inconsistency. The correct process should indeed involve subtracting 50 from both sides of the equation. Let's correct that step.

### Step 2: Subtract 50 from Both Sides

We want to find when the percent of kernels popping \( P \) is at least 50. So, we set up the inequality as:

\[
-\frac{1}{250}T^2 + 2.8T - 394 \geq 50
\]

Next, subtract 50 from both sides to get:

\[
-\frac{1}{250}T^2 + 2.8T - 444 \geq 0
\]

The subtraction is necessary to bring all terms to one side, leaving zero on the other side, setting up the inequality for solving.

The remaining steps, such as multiplying by -250 and solving the quadratic equation, were carried out correctly.

### Corrected Answer

The correct temperature range that ensures at least 50% of the kernels will pop is **242.8°F to 457.3°F**.

Would you like more details or have any other questions? Here are five questions that could be explored further:

1. How does multiplying an inequality by a negative number affect the inequality sign?
2. Why is it important to get the quadratic inequality in standard form before solving?
3. What role does the vertex of the parabola play in determining the temperature range?
4. How would the inequality change if the temperature units were in Celsius?
5. What is the significance of the temperature endpoints in practical terms for popcorn popping?

**Tip:** Always double-check each arithmetic step in solving equations to ensure consistency, especially when handling signs.

Understand the correct arithmetic operations in solving quadratic inequalities step-by-step. This problem involves setting up and solving a quadratic inequality with detailed explanations on subtracting and setting up the inequality correctly. Suitable for high school students.

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