Let's break down the problem:

Cecilia starts with $10 in her bank account, and every week she deposits $50. We want to determine how many weeks of deposits would result in her account balance being at least $410.

Step 1: Express the situation mathematically.

After $n$ weeks, Cecilia will have deposited $50n$ dollars in addition to her initial $10. Therefore, her total balance after $n$ weeks is:

$\text{Balance} = 50n + 10$

We know that her balance is at least $410, so we can write the inequality:

$50n + 10 \geq 410$

Step 2: Solve the inequality.

Subtract 10 from both sides:

$50n \geq 400$

Now, divide both sides by 50:

$n \geq 8$

Conclusion:

Cecilia has been depositing money for at least 8 weeks for her balance to reach or exceed $410.

Would you like more details or explanations on any step?

Here are 5 related questions:

1. What would Cecilia’s balance be after 12 weeks of deposits?
2. How much money does she need to deposit to reach $1,000?
3. If she stops depositing money after 10 weeks, what will her balance be?
4. If she withdraws $20 every 3 weeks, how does that affect the balance?
5. How much time would it take her to double her balance after reaching $410?

Tip: To solve inequalities involving word problems, first translate the scenario into a mathematical expression and then solve as you would with equations.