Solve the inequality and graph the solution on the number line. $$ 3 x+6

Question

Solve the inequality and graph the solution on the number line. $$ 3 x+6<2 x+8 $$ or $$ 2 x+4 \leq 4 x-6 $$

UpStudy AI · Accepted Answer

To solve the inequality and graph the solution on the number line, we'll tackle each part of the compound inequality separately and then combine the results.

### Given Inequality:
$$ 3x + 6 < 2x + 8 \quad \text{or} \quad 2x + 4 \leq 4x - 6 $$

### Solving Each Inequality:

1. **First Inequality:**
   $$ 3x + 6 < 2x + 8 $$
   
   - **Subtract $$2x$$ from both sides:**
     $$ 3x - 2x + 6 < 8 $$
     $$ x + 6 < 8 $$
   
   - **Subtract 6 from both sides:**
     $$ x < 2 $$

**Solution:** $$ x < 2 $$

2. **Second Inequality:**
   $$ 2x + 4 \leq 4x - 6 $$
   
   - **Subtract $$2x$$ from both sides:**
     $$ 4 \leq 2x - 6 $$
   
   - **Add 6 to both sides:**
     $$ 10 \leq 2x $$
   
   - **Divide both sides by 2:**
     $$ 5 \leq x $$
     $$ x \geq 5 $$

**Solution:** $$ x \geq 5 $$

### Combining the Solutions:
The original compound inequality uses "or," meaning the solution is the union of both individual solutions.

$$ x < 2 \quad \text{or} \quad x \geq 5 $$

### Graphing on the Number Line:

1. **For $$ x < 2 $$:**
   - Draw an **open circle** at $$ x = 2 $$ (since 2 is not included).
   - Shade the line to the **left** of 2.

2. **For $$ x \geq 5 $$:**
   - Draw a **closed circle** at $$ x = 5 $$ (since 5 is included).
   - Shade the line to the **right** of 5.

### Number Line Representation:

```
<---|-----------|===========>
    2           5
```
- **Open circle** at 2 (indicated by the parenthesis in the interval).
- **Closed circle** at 5 (indicated by the square bracket in the interval).
- Shading extends to the left of 2 and to the right of 5.

### Final Solution:
$$ x < 2 \quad \text{or} \quad x \geq 5 $$

### Interval Notation:
$$ (-\infty, 2) \cup [5, \infty) $$
The solution to the inequality $$ 3x + 6 < 2x + 8 $$ or $$ 2x + 4 \leq 4x - 6 $$ is $$ x < 2 $$ or $$ x \geq 5 $$. On the number line, this is represented by an open circle at 2 and shading to the left, and a closed circle at 5 with shading to the right.

Solve the inequality and graph the solution on the number line. \( 3 x+6<2 x+8 \) or \( 2 x+4 \leq 4 x-6 \)

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