Solve and graph the solution set of 3x + 6 `ge 9 and - 5x gt - 15`, where `x in R`.

Answer

To solve the inequalities \(3x + 6 \geq 9\) and \(-5x > -15\), we will follow these steps:

### Step 1: Solve the first inequality \(3x + 6 \geq 9\)

1. Subtract 6 from both sides:
\[ 3x + 6 - 6 \geq 9 - 6 \]
This simplifies to:
\[ 3x \geq 3 \]

2. Divide both sides by 3:
\[ x \geq 1 \]

### Step 2: Solve the second inequality \(-5x > -15\)

1. Divide both sides by -5. Remember, when dividing by a negative number, we must reverse the inequality sign:
\[ x < 3 \]

### Step 3: Combine the results

From Step 1, we have \(x \geq 1\) and from Step 2, we have \(x < 3\). Therefore, the combined solution is:
\[ 1 \leq x < 3 \]

### Step 4: Graph the solution set

1. Draw a number line.
2. Mark the point 1 with a closed dot (since \(x\) can be equal to 1).
3. Mark the point 3 with an open dot (since \(x\) cannot be equal to 3).
4. Shade the region between 1 and 3 to indicate that \(x\) can take any value in this interval.

### Final Answer

The solution set is:
\[ [1, 3) \]

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