### Step 1: Understanding the Problem

Rob has two different pay rates for two different jobs at the riding stable:

- $5 per hour for exercising horses.
- $10 per hour for cleaning stalls.

He is a full-time student, so that he can work up to 12 hours per week. His goal is to earn at least $60 per week.

### Step 2: Defining the Constraints

Rob’s work constraints can be translated into the following inequalities:

- Total hours worked per week (
*e*+*c*) must be less than or equal to 12 hours:*e*+*c*≤ 12. - Total earnings per week (
*e*+ 10*c*) must be at least $60: 5*e*+ 10*c*≥ 60.

Where:

*e*= hours spent exercising horses.*c*= hours spent cleaning stalls.

### Step 3: Evaluating the Options

Rob has four potential work schedules to consider:

#### Option A: 7 hours exercising and 8 hours cleaning

Total hours: 7 + 8 = 15 hours

Total earnings: 5 × 7 + 10 × 8 = 35 + 80 = 115 dollars

#### Option B: 1 hour exercising and 1 hour cleaning

Total hours: 1 + 1 = 2 hours

Total earnings: 5 × 1 + 10 × 1 = 5 + 10 = 15 dollars

#### Option C: 2 hours exercising and 8 hours cleaning

Total hours: 2 + 8 = 102 + 8 = 10 hours

Total earnings: 5 × 2 + 10 × 8 = 10 + 80 = 90 dollars

#### Option D: 2 hours exercising and 5 hours cleaning

Total hours: 2 + 5 = 7 hours

Total earnings: 5 × 2 + 10 × 5 = 10 + 50 = 60 dollars

### Step 4: Checking Each Option Against the Constraints

Now, we’ll check which options meet both the hours constraint (≤ 12 hours) and the pay constraint (≥ 60 dollars).

**Option A**fails the hours constraint as Rob would work 15 hours, more than the 12-hour limit.**Option B**fails the pay constraint as Rob would only earn $15 below the $60 requirement.**Option C**meets both constraints, as Rob would work 10 hours and earn $90.**Option D**also meets both constraints, with 7 hours of work and $60 in earnings.

### Step 5: Determining the Correct Option

After evaluating all the options, we can see that:

**Option C (2 hours exercising and 8 hours cleaning)**is a valid solution because it meets both constraints: Rob works 10 hours, which is under the 12-hour limit, and he earns $90, which is above the $60 minimum.**Option D (2 hours exercising and 5 hours of cleaning)**is also a valid solution because it meets both constraints: Rob works 7 hours and earns exactly $60.

Therefore, both Options C and D are possible solutions for Rob’s work schedule based on the system of inequalities provided.