Solve the inequality, then graph the solution set. $$ |x-2| \geq 4 $$

Question

UpStudy AI · Accepted Answer

To solve the inequality $$ |x - 2| \geq 4 $$ and graph the solution set, follow these steps:

### Step 1: Understand the Absolute Value Inequality

The inequality $$ |x - 2| \geq 4 $$ represents all real numbers $$ x $$ whose distance from 2 is at least 4 units on the number line.

### Step 2: Split the Inequality

An absolute value inequality of the form $$ |A| \geq B $$ (where $$ B \geq 0 $$) can be rewritten as two separate inequalities:
1. $$ A \geq B $$
2. $$ A \leq -B $$

Applying this to $$ |x - 2| \geq 4 $$:
1. $$ x - 2 \geq 4 $$
2. $$ x - 2 \leq -4 $$

### Step 3: Solve Each Inequality Separately

1. **First Inequality:**
   $$
   x - 2 \geq 4 \
   x \geq 4 + 2 \
   x \geq 6
   $$

2. **Second Inequality:**
   $$
   x - 2 \leq -4 \
   x \leq -4 + 2 \
   x \leq -2
   $$

### Step 4: Combine the Solutions

The solutions to the inequality $$ |x - 2| \geq 4 $$ are all real numbers $$ x $$ such that:
$$
x \leq -2 \quad \text{or} \quad x \geq 6
$$

In **interval notation**, this is written as:
$$
(-\infty, \, -2] \cup [6, \, \infty)
$$

### Step 5: Graph the Solution Set

Below is a graph representing the solution set on a number line:

```
<----|=================•-------------------•=================|---->
    -∞               -2                   6                 ∞
```

- **Closed Dots at -2 and 6:** Indicate that -2 and 6 are included in the solution (≥).
- **Shaded Arrows:**
  - To the **left of -2**: Represents all $$ x $$ values less than or equal to -2.
  - To the **right of 6**: Represents all $$ x $$ values greater than or equal to 6.

### Visual Representation

For a clearer visual, here's a number line illustration:

![Graph of |x-2| ≥ 4](https://i.imgur.com/YourImageLink.png)

*(Note: Replace the image link with an actual graph if you're presenting this in a medium that supports images.)*

### Summary

- **Solution:** $$ x \leq -2 $$ or $$ x \geq 6 $$
- **Interval Notation:** $$ (-\infty, \, -2] \cup [6, \, \infty) $$
- **Graph:** Number line with shaded regions extending to the left of -2 and to the right of 6, including the points -2 and 6.
The solution to the inequality $$ |x - 2| \geq 4 $$ is all real numbers $$ x $$ such that $$ x \leq -2 $$ or $$ x \geq 6 $$. In interval notation, this is $$ (-\infty, \, -2] \cup [6, \, \infty) $$.

Solve the inequality, then graph the solution set.

Upstudy AI Solution

Answer

Solution

Step 1: Understand the Absolute Value Inequality

Step 2: Split the Inequality

Step 3: Solve Each Inequality Separately

Step 4: Combine the Solutions

Step 5: Graph the Solution Set

Visual Representation

Summary

Mind Expander

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