Let's break down the given problem step by step.

We are told:

1. The first hose fills at a rate of x gallons per minute.
2. The second hose fills at a rate of 15 gallons per minute less than the first hose. This means the second hose fills at a rate of x - 15 gallons per minute.
3. The combined rate of the two hoses must be at least 10 gallons per minute. This means the total rate is represented by the inequality:

$x + (x - 15) \geq 10$

Now, we can solve this inequality.

Step 1: Simplify the inequality

$x + x - 15 \geq 10$ $2x - 15 \geq 10$

Step 2: Solve for x

Add 15 to both sides: $2x \geq 25$

Now, divide both sides by 2: $x \geq \frac{25}{2} = 12.5$

Step 3: Interpret the solution

The inequality $x \geq 12.5$ tells us that for the combined rate of the two hoses to be at least 10 gallons per minute, the rate of the first hose (x) must be at least 12.5 gallons per minute.

Conclusion

The answer choice that represents this inequality would be the one that includes all values of x that are 12.5 or greater.

So, the correct answer will likely be one of these:

• "Both intervals contain viable rates for the hoses"
• "Only mc027-7.jpg contains viable rates for the hoses"

But to be certain, you would need to examine the graph or diagram referred to as mc027-1.jpg, mc027-2.jpg, etc., to match the values visually.

Would you like me to look at a graph for you, or would you like further clarification on this solution?