Solve and graph the solution set of `-2 lt 2x - 6 or -2x + 5 ge 13`. Where `x in R`.

Answer

To solve the inequality \(-2 < 2x - 6\) or \(-2x + 5 \geq 13\), we will break it down into two separate inequalities and solve each one step by step.

### Step 1: Solve the first inequality \(-2 < 2x - 6\)

1. **Add 6 to both sides**:
\[ -2 + 6 < 2x \]
\[ 4 < 2x \]

2. **Divide both sides by 2**:
\[ \frac{4}{2} < x \]
\[ 2 < x \quad \text{or} \quad x > 2 \]

### Step 2: Solve the second inequality \(-2x + 5 \geq 13\)

1. **Subtract 5 from both sides**:
\[ -2x \geq 13 - 5 \]
\[ -2x \geq 8 \]

2. **Divide both sides by -2** (remember to flip the inequality sign):
\[ x \leq \frac{8}{-2} \]
\[ x \leq -4 \]

### Step 3: Combine the solutions

From the two inequalities, we have:
- From the first inequality: \(x > 2\)
- From the second inequality: \(x \leq -4\)

Since the question uses "or," the solution set is:
\[ x > 2 \quad \text{or} \quad x \leq -4 \]

### Step 4: Graph the solution set

1. **Draw a number line**.
2. **Mark the points**:
- Open circle at \(2\) (since \(x > 2\))
- Closed circle at \(-4\) (since \(x \leq -4\))

3. **Shade the regions**:
- Shade to the right of \(2\) (indicating all values greater than \(2\)).
- Shade to the left of \(-4\) (indicating all values less than or equal to \(-4\)).

### Final Solution

The solution set is:
\[ x > 2 \quad \text{or} \quad x \leq -4 \]