Tessa is attempting to lose weight by counting the number of calories she consumes. Her goal is to eat less than 2,000 calories a day. She can burn calories through exercise. Each minute she runs on the treadmill burns 12.5 calories. Today she ate 2,500 calories of food. How many minutes must she run to burn enough to be within her daily calorie limit?

Write and solve an inequality to represent the situation. Then, graph the solution set on a number line and verify. Finally, interpret the solution in terms of the problem.

### Step 1: Write the Inequality

Tessa's goal is to eat fewer than 2,000 net calories for the day. She starts with 2,500 calories consumed and burns 12.5 calories per minute of treadmill running. The inequality representing this situation is:

\[
2500 - 12.5x < 2000
\]

where \(x\) is the number of minutes she needs to run.

---

### Step 2: Solve the Inequality

1. Subtract 2,000 from both sides:
   \[
   2500 - 12.5x - 2000 < 0
   \]

   Simplify:
   \[
   500 - 12.5x < 0
   \]

2. Subtract 500 from both sides:
   \[
   -12.5x < -500
   \]

3. Divide both sides by \(-12.5\) and reverse the inequality sign:
   \[
   x > 40
   \]

---

### Step 3: Graph the Solution Set

The solution \(x > 40\) means Tessa must run **more than 40 minutes**. On a number line:

1. Draw a line representing the range of possible \(x\) values (minutes).
2. Mark \(40\) with an open circle (because it is not included in the solution).
3. Shade to the right of \(40\), indicating \(x > 40\).

---

### Step 4: Verify the Solution

Let’s check a value greater than 40, say \(x = 50\):
\[
2500 - 12.5(50) = 2500 - 625 = 1875
\]
Since \(1875 < 2000\), this satisfies the inequality.

Now, check \(x = 40\):
\[
2500 - 12.5(40) = 2500 - 500 = 2000
\]
Since \(2000 \not< 2000\), \(x = 40\) is not part of the solution.

---

### Step 5: Interpret the Solution

Tessa must run for more than **40 minutes** on the treadmill to burn enough calories to achieve her goal of staying under 2,000 net calories for the day.

---

### Additional Questions for Expansion

1. How many calories would Tessa burn if she ran for 60 minutes?
2. What is the inequality if Tessa's goal changes to eating fewer than 1,800 calories net per day?
3. How does the calorie burn rate change the number of minutes she needs to run?
4. If Tessa ate 3,000 calories instead, how many minutes would she need to run?
5. What is the graph of the inequality \(2500 - 12.5x \leq 2000\)?

---

### Tip
Always double-check your inequality by testing values within and outside the solution set to ensure the correct interpretation.

Tessa is attempting to lose weight by counting the number of calories she consumes. Her goal is to eat less than 2,000 calories a day. She can burn calories through exercise. Each minute she runs on the treadmill burns 12.5 calories. Today she ate 2,500 calories of food. How many minutes must she run to burn enough to be within her daily calorie limit?

Explore how to solve a real-world inequality where Tessa aims to stay under 2,000 net calories by running on a treadmill. This problem involves solving the inequality 2500 - 12.5x < 2000 and interpreting the solution. Perfect for Grades 6-8.

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