The question in your image states: "When 2 times a number is subtracted from 4, the absolute value of the difference is at least 6. Use interval notation to express the set of all numbers that satisfy this condition."

Let's break it down and solve the problem step-by-step.

Step 1: Set up the equation

We are asked for the absolute value of $4 - 2x$, where $x$ is a number, and that the absolute value is at least 6. In mathematical terms, this is written as:

$|4 - 2x| \geq 6$

Step 2: Solve the absolute value inequality

An absolute value inequality like $|A| \geq B$ means that either:

1. $A \geq B$, or
2. $A \leq -B$

Apply this to our equation:

$4 - 2x \geq 6 \quad \text{or} \quad 4 - 2x \leq -6$

Step 3: Solve each inequality

Inequality 1: $4 - 2x \geq 6$

Subtract 4 from both sides: $-2x \geq 2$ Now, divide by -2 (remember to reverse the inequality): $x \leq -1$

Inequality 2: $4 - 2x \leq -6$

Subtract 4 from both sides: $-2x \leq -10$ Now, divide by -2 (again, reverse the inequality): $x \geq 5$

Step 4: Write the solution in interval notation

The solution is $x \leq -1$ or $x \geq 5$. In interval notation, this is:

$(-\infty, -1] \cup [5, \infty)$

Final Answer:

The solution set is: $(-\infty, -1] \cup [5, \infty)$

Would you like more details on any part of this solution?

Here are 5 related questions to deepen your understanding:

1. How do we solve inequalities involving absolute values in general?
2. What are the rules for solving inequalities when dividing by a negative number?
3. Can you solve the inequality $|3x - 7| \geq 5$?
4. What is the difference between $|A| \geq B$ and $|A| \leq B$?
5. How can we visualize solutions to absolute value inequalities on a number line?

Tip: Always remember to reverse the inequality sign when dividing by a negative number during inequality solving.