The image shows a number line with several points labeled $a$, $0$, and $b$. The positions of $a$ and $b$ relative to $0$ suggest the following relationships:

• $a$ is to the left of 0, so $a < 0$.
• $b$ is to the right of 0, so $b > 0$.

The question asks which of the following inequalities are correct:

1. $|a| < 1$
2. $b < 1$
3. $|b| = -1$
4. $b > -1$

Analysis of Each Statement:

1. $|a| < 1$: This suggests that the absolute value of $a$ is less than 1, meaning $a$ is between -1 and 0. Based on the number line, $a$ appears to lie between $-1$ and $0$, so this inequality is true.

2. $b < 1$: This suggests that $b$ is less than 1. Looking at the number line, $b$ appears to be less than 1, so this statement is true.

3. $|b| = -1$: This is a false statement, as the absolute value of a number is always non-negative. Absolute values cannot be equal to a negative number, so this is false.

4. $b > -1$: Since $b$ is positive (greater than 0), this inequality is true.

Correct Answers:

The correct inequalities are:

• $|a| < 1$
• $b < 1$
• $b > -1$

Do you want any further details or have any questions?

Here are five related questions:

1. What is the definition of absolute value, and why can't it be negative?
2. How do we determine the position of a point on a number line using inequalities?
3. Why is $|b| = -1$ always false, regardless of the value of $b$?
4. How would these inequalities change if $a$ or $b$ were on the other side of 0?
5. Can we use number lines to represent inequalities involving more than two variables?

Tip: Always check for basic properties of mathematical operations (like absolute value) when solving inequalities.