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The problem states that:

- Julie has already decorated 53 cookies and can decorate 10 cookies per hour.
- Amy has already decorated 25 cookies and can decorate 14 cookies per hour.

We are asked to find how many hours it will take for Amy to have decorated at least as many cookies as Julie. Let \( x \) represent the number of hours they spend decorating cookies.

### Step 1: Write the inequality
To model this situation, we can create an equation for the total number of cookies each person decorates after \( x \) hours.

- Julie's cookies after \( x \) hours: \( 53 + 10x \)
- Amy's cookies after \( x \) hours: \( 25 + 14x \)

We need to find when Amy’s cookies are at least equal to Julie’s cookies, so the inequality becomes:

\[
25 + 14x \geq 53 + 10x
\]

### Step 2: Solve the inequality
Now, let's solve this inequality:

\[
25 + 14x \geq 53 + 10x
\]

First, subtract \( 10x \) from both sides:

\[
25 + 4x \geq 53
\]

Next, subtract 25 from both sides:

\[
4x \geq 28
\]

Finally, divide by 4:

\[
x \geq 7
\]

### Conclusion:
The inequality that represents this situation is \( 25 + 14x \geq 53 + 10x \), and Amy will have decorated at least as many cookies as Julie after **7 hours**.

---

Would you like further clarification on any steps? Here are 5 follow-up questions that build on this topic:

1. How would the inequality change if Julie decorated 12 cookies per hour instead of 10?
2. If they both stopped after 5 hours, how many cookies would each have decorated?
3. How would the equation change if Amy had decorated 30 cookies initially instead of 25?
4. What happens if Julie decorated 60 cookies initially? How does this impact the number of hours needed for Amy to catch up?
5. Can you think of a real-life scenario where inequalities are used to compare two different rates like in this cookie-decorating problem?

**Tip:** When solving inequalities, remember that if you multiply or divide by a negative number, the inequality sign flips!

Julie and Amy are decorating cookies. Julie has decorated 53 cookies and can decorate 10 cookies every hour. Amy has decorated 25 cookies and can decorate 14 cookies every hour. After how many hours will Amy have decorated at least as many cookies as Julie?

Learn how to solve an inequality problem comparing the cookie decorating speeds of Julie and Amy. This problem demonstrates how to set up and solve a linear inequality to find when Amy will have decorated at least as many cookies as Julie. Suitable for students in Grades 7-9.

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