https://file.aiquickdraw.com/mathbot/file/1725461120384u2mw8dko.jpg

The image shows a series of inequalities involving logarithms that need to be solved. The inequalities are:

1. \(\log(x) + 1 \leq 2\)
2. \(\log(x) + 1 \geq 2\)
3. \(\log(x) + 1 \leq -2\)
4. \(\log(x) + 1 \geq -2\)

Let's solve each inequality:

### 1. \(\log(x) + 1 \leq 2\)

\[
\log(x) \leq 2 - 1 = 1
\]

To remove the logarithm, raise both sides as powers of 10:

\[
x \leq 10^1 = 10
\]

The solution is \(x \leq 10\).

### 2. \(\log(x) + 1 \geq 2\)

\[
\log(x) \geq 2 - 1 = 1
\]

Again, raise both sides as powers of 10:

\[
x \geq 10^1 = 10
\]

The solution is \(x \geq 10\).

### 3. \(\log(x) + 1 \leq -2\)

\[
\log(x) \leq -2 - 1 = -3
\]

Raise both sides as powers of 10:

\[
x \leq 10^{-3} = 0.001
\]

However, since \(\log(x)\) is only defined for \(x > 0\), the solution is \(0 < x \leq 0.001\).

### 4. \(\log(x) + 1 \geq -2\)

\[
\log(x) \geq -2 - 1 = -3
\]

Raise both sides as powers of 10:

\[
x \geq 10^{-3} = 0.001
\]

The solution is \(x \geq 0.001\).

### Summary of Solutions:
- Inequality 1: \(x \leq 10\)
- Inequality 2: \(x \geq 10\)
- Inequality 3: \(0 < x \leq 0.001\)
- Inequality 4: \(x \geq 0.001\)

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you solve inequalities involving natural logarithms?
2. What are the domain restrictions for logarithmic functions?
3. How do you interpret the intersection of solutions for multiple inequalities?
4. How does changing the base of the logarithm affect the solution?
5. What is the significance of inequalities in logarithmic functions in real-world applications?

**Tip:** When dealing with logarithmic inequalities, always consider the domain restrictions, as logarithms are only defined for positive arguments.

Solve the following logarithmic inequalities:
1. log(x) + 1 ≤ 2
2. log(x) + 1 ≥ 2
3. log(x) + 1 ≤ -2
4. log(x) + 1 ≥ -2

This webpage provides a detailed step-by-step solution for solving the logarithmic inequalities: log(x) + 1 ≤ 2, log(x) + 1 ≥ 2, log(x) + 1 ≤ -2, and log(x) + 1 ≥ -2. It covers key concepts in algebra and logarithmic functions, suitable for students in Grades 10-12.

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