Solve `-3le3x + 2<4`, `xinR`.

Answer

To solve the inequality \(-3 \leq 3x + 2 < 4\), we will break it down into two parts and solve for \(x\).

### Step 1: Break down the compound inequality
We can separate the compound inequality into two parts:
1. \(-3 \leq 3x + 2\)
2. \(3x + 2 < 4\)

### Step 2: Solve the first part \(-3 \leq 3x + 2\)
To isolate \(3x\), we subtract \(2\) from both sides:
\[ -3 - 2 \leq 3x \]
This simplifies to:
\[ -5 \leq 3x \]
Now, divide both sides by \(3\):
\[ -\frac{5}{3} \leq x \]

### Step 3: Solve the second part \(3x + 2 < 4\)
Again, we isolate \(3x\) by subtracting \(2\) from both sides:
\[ 3x < 4 - 2 \]
This simplifies to:
\[ 3x < 2 \]
Now, divide both sides by \(3\):
\[ x < \frac{2}{3} \]

### Step 4: Combine the results
Now we have two inequalities:
\[ -\frac{5}{3} \leq x < \frac{2}{3} \]
This means that \(x\) lies in the interval:
\[ x \in \left[-\frac{5}{3}, \frac{2}{3}\right) \]

### Final Answer
Thus, the solution set for \(x\) is:
\[ x \in \left[-\frac{5}{3}, \frac{2}{3}\right) \]

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