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The given table shows an estimate for the solution to the equation \(5x + 5 = 3 \cdot 2^x\), by using different values of \(x\).

### Equation:
\[ 5x + 5 = 3 \cdot 2^x \]

### Table Overview:

| \(x\)  | \(5x + 5\) | \(3 \cdot 2^x\) |
|-------|------------|----------------|
| 1     | 10         | 6              |
| 2     | 15         | 12             |
| 3     | 20         | 24             |
| 2.5   | 17.5       | 17.0           |
| 2.6   | 18         | 18.2           |
| 2.55  | 17.8       | 17.6           |

We are asked to estimate the solution where \(5x + 5 \approx 3 \cdot 2^x\), meaning we are looking for the point where the two sides are approximately equal.

### Observing the table:
- At \(x = 2.5\), we have:
  \[
  5x + 5 = 17.5 \quad \text{and} \quad 3 \cdot 2^x = 17.0 \quad (\text{Difference: } 0.5)
  \]
- At \(x = 2.6\), we have:
  \[
  5x + 5 = 18 \quad \text{and} \quad 3 \cdot 2^x = 18.2 \quad (\text{Difference: } 0.2)
  \]
- At \(x = 2.55\), we have:
  \[
  5x + 5 = 17.8 \quad \text{and} \quad 3 \cdot 2^x = 17.6 \quad (\text{Difference: } 0.2)
  \]

From the table, we can see that both \(x = 2.6\) and \(x = 2.55\) provide very close estimates, with a small difference of 0.2 on each side.

### Estimated Solution:
Therefore, to the nearest tenth, the solution is approximately:
\[ x \approx 2.6 \]

Would you like more details or explanations?

Here are five related questions:
1. How would you solve this equation algebraically?
2. How does the exponential term \(2^x\) affect the solution compared to a linear equation?
3. What methods are used to estimate solutions for equations without direct algebraic solutions?
4. How would graphing both sides of the equation help in finding the solution?
5. Why do we round to the nearest tenth in this scenario?

**Tip:** For exponential equations like this, using a combination of estimation and graphing tools can often help visualize where the two sides intersect.

Use the table to estimate one solution, to the nearest tenth, of the equation: 5x + 5 = 3 * 2^x.

Learn how to estimate the solution to the exponential equation 5x + 5 = 3 * 2^x by using a table of values. This problem demonstrates the process of estimation through observation of approximate equality at specific values of x. Suitable for students in Grades 9-11.

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