Tom can work no more than 25 hours per week, job 1 pays 5 dollars, job 2 pays 10 dollars, he needs to make at least 175 dollars

Let's break this down:

- Tom can work at most **25 hours per week**.
- **Job 1** pays **$5 per hour**.
- **Job 2** pays **$10 per hour**.
- Tom needs to make at least **$175 per week**.

Let:
- $ x $ be the number of hours Tom works at Job 1.
- $ y $ be the number of hours Tom works at Job 2.

### Constraints:
1. $ x + y \leq 25 $ (since he can work no more than 25 hours per week).
2. $ 5x + 10y \geq 175 $ (since he needs to make at least $175).
3. $ x \geq 0 $ and $ y \geq 0 $ (he cannot work negative hours).

### Solving the System:
The system of inequalities is:
\[
\begin{aligned}
    x + y & \leq 25, \\
    5x + 10y & \geq 175, \\
    x & \geq 0, \\
    y & \geq 0.
\end{aligned}
\]

### Step 1: Simplify the second inequality
We can divide the second inequality $ 5x + 10y \geq 175 $ by 5:
\[
x + 2y \geq 35.
\]

### Step 2: Analyze the system of inequalities
The system becomes:
\[
\begin{aligned}
    x + y & \leq 25, \\
    x + 2y & \geq 35, \\
    x & \geq 0, \\
    y & \geq 0.
\end{aligned}
\]

### Step 3: Solve the system graphically or algebraically
We can now solve for the values of $ x $ and $ y $ that satisfy both inequalities.

- From $ x + y \leq 25 $, we get $ y \leq 25 - x $.
- From $ x + 2y \geq 35 $, we get $ y \geq \frac{35 - x}{2} $.

To find feasible solutions, we look for values of $ x $ and $ y $ that satisfy both these conditions.

Would you like me to solve it further by finding specific values for $ x $ and $ y $? 

Here are some questions related to this scenario:

1. What would happen if Tom wanted to make more than $175 per week?
2. How would the solution change if Job 1 paid $8 per hour instead?
3. Can Tom work fewer hours if the wage at Job 2 increases? How?
4. How many hours would Tom need to work if he only worked Job 2 to make $175?
5. What is the maximum amount of money Tom can make in a week if he works the full 25 hours?

**Tip**: In problems with multiple constraints, graphing inequalities can help you visualize the solution set.

Explore how to solve a real-world linear inequalities problem where Tom needs to allocate his work hours between two jobs to earn at least $175 per week. Learn about solving systems of inequalities, simplifying constraints, and finding the feasible region. Ideal for high school students in Grades 9-12.

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