To determine which inequality represents the number of weeks [tex]\( w \)[/tex] that Manuel can withdraw money while maintaining a balance of at least \[tex]$300, we need to set up and solve the inequality step by step. Here is the detailed solution:
1. Understand the Problem:
- Initial savings: \$[/tex]600
- Minimum desired savings: \[tex]$300
- Weekly withdrawal amount: \$[/tex]28
2. Establish the Inequality:
Manuel wants to ensure that his savings do not drop below \[tex]$300. If he withdraws \$[/tex]28 each week for [tex]\( w \)[/tex] weeks, the amount left in his account after [tex]\( w \)[/tex] weeks can be represented as:
[tex]\[
\text{Remaining balance} = 600 - 28w
\][/tex]
This remaining balance should be at least \[tex]$300:
\[
600 - 28w \geq 300
\]
3. Solve the Inequality:
We solve for \( w \) by isolating it on one side of the inequality:
\[
600 - 28w \geq 300
\]
Subtract 300 from both sides:
\[
600 - 300 - 28w \geq 0
\]
Simplify the left-hand side:
\[
300 - 28w \geq 0
\]
Rearrange the inequality to solve for \( w \):
\[
300 \geq 28w
\]
Divide both sides by 28:
\[
\frac{300}{28} \geq w
\]
Simplify the fraction:
\[
10.714285714285714 \geq w
\]
This inequality tells us that \( w \) can be a maximum of approximately 10.71 weeks. Since \( w \) must be an integer (he withdraws each week), the largest whole number \( w \) that satisfies the inequality is 10 weeks.
4. Conclusion:
The inequality that represents the number of weeks Manuel can withdraw money while not dropping below a \$[/tex]300 balance is:
[tex]\[
600 - 28w \geq 300
\][/tex]
Therefore, the correct answer is:
[tex]\[
600 - 28 w \geq 300
\][/tex]
