The inequality in the image is:

$1 \leq |x| < 4$

To solve this inequality, we can break it down as follows:

Step 1: Understanding the absolute value inequality

The absolute value inequality $|x|$ describes the distance of $x$ from 0. Specifically:

• $|x| \geq 1$ means that $x$ is at least 1 unit away from 0, in either the positive or negative direction.
• $|x| < 4$ means that $x$ is less than 4 units away from 0, in either the positive or negative direction.

Step 2: Solving the inequality

The inequality can be broken into two cases for each part:

$1 \leq |x| < 4$

This translates to two simultaneous inequalities:

$1 \leq |x|$ and $|x| < 4$

For $|x| \geq 1$:

• This means $x \leq -1$ or $x \geq 1$ (since the absolute value is greater than or equal to 1).

For $|x| < 4$:

• This means $-4 < x < 4$.

Step 3: Combine the two conditions

Now, we combine the results from the two inequalities:

1. $1 \leq |x|$ gives us $x \leq -1$ or $x \geq 1$.
2. $|x| < 4$ gives us $-4 < x < 4$.

Combining these two conditions, we get:

• For the first part $x \leq -1$, we need $-4 < x \leq -1$, so $-4 < x \leq -1$.
• For the second part $x \geq 1$, we need $1 \leq x < 4$.

Final Answer:

The solution to the inequality is:

$-4 < x \leq -1 \quad \text{or} \quad 1 \leq x < 4$

This is the set of all $x$ such that $x$ is between -4 and -1 (inclusive of -1) or between 1 and 4 (inclusive of 1, but not 4).

Would you like more details on how to solve inequalities involving absolute values?

Here are some related questions for practice:

1. What is the general process for solving absolute value inequalities?
2. How would the solution change if the inequality were $|x| \geq 4$?
3. Can you graph the solution on a number line?
4. What is the difference between $|x| < a$ and $|x| \leq a$?
5. How do you solve the inequality $|x - 2| < 5$?

Tip: When solving absolute value inequalities, always consider both the positive and negative cases for the absolute value function.