solve a: (1 point) Let S be the union of the two intervals (−∞,−4]
 and [4,∞)
.
Then S can also be described by the inequality |x−a|≥b

The problem asks to describe the union of the intervals \( (-\infty, -4] \) and \( [4, \infty) \) using an absolute value inequality of the form \( |x - a| \geq b \).

### Step 1: Understand the intervals
The union of the intervals \( (-\infty, -4] \) and \( [4, \infty) \) includes all values of \( x \) that are either:
- Less than or equal to \(-4\), or
- Greater than or equal to \(4\).

This can be interpreted geometrically as the set of points that are at least a distance of 4 units away from 0 on the real number line.

### Step 2: Set up the absolute value inequality
An absolute value inequality of the form \( |x - a| \geq b \) describes all points that are at least \( b \) units away from \( a \). In our case:
- The points are at least 4 units away from 0 (since \(-4\) and \(4\) are symmetric around 0).

So, we have:
\[
|x - 0| \geq 4
\]
which simplifies to:
\[
|x| \geq 4
\]

### Conclusion:
The union of the intervals \( (-\infty, -4] \cup [4, \infty) \) can be described by the inequality \( |x| \geq 4 \).

### Final form:
\[
|x - 0| \geq 4 \quad \text{or simply} \quad |x| \geq 4
\]

Would you like further details on this explanation or how this concept works?

### Related Questions:
1. How would you describe \( (-\infty, -3] \cup [3, \infty) \) as an absolute value inequality?
2. What does the inequality \( |x - 2| \geq 5 \) represent geometrically?
3. How can you represent the interval \( [-5, 5] \) using an absolute value inequality?
4. What is the general formula for expressing intervals as absolute value inequalities?
5. How would you represent the set \( (-\infty, -7] \cup [7, \infty) \) with an absolute value inequality?

### Tip:
Absolute value inequalities like \( |x - a| \geq b \) describe points that are at least \( b \) units away from \( a \) on the real number line.

Let S be the union of the two intervals (-∞, -4] and [4, ∞). Then S can also be described by the inequality |x−a|≥b.

Understand how to represent the union of intervals (-∞, -4] and [4, ∞) using an absolute value inequality |x| ≥ 4. This detailed step-by-step explanation covers key algebraic concepts, including intervals and absolute value inequalities. Suitable for Grades 9-11.

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