mutlak 3 x min 1 kurang dari 2 mutlak x + 6

To solve the inequality involving absolute values given in your question, we can rewrite it in mathematical terms for clarity:

\[
|3x - 1| < 2|x + 6|
\]

### Step 1: Understand the Absolute Values
Absolute values indicate the distance of a number from zero, so they are always non-negative. To handle this inequality, consider the different cases based on the expression inside the absolute value.

### Step 2: Set Up the Cases
There are different cases to consider based on the points where the expressions inside the absolute values change sign:
- \(3x - 1 = 0\) gives \(x = \frac{1}{3}\)
- \(x + 6 = 0\) gives \(x = -6\)

These points split the number line into three intervals:
1. \(x < -6\)
2. \(-6 \leq x < \frac{1}{3}\)
3. \(x \geq \frac{1}{3}\)

### Step 3: Solve Each Case

#### Case 1: \(x < -6\)
For this range, \(3x - 1 < 0\) and \(x + 6 < 0\), so the absolute values can be rewritten as:
\[
|3x - 1| = -(3x - 1) \quad \text{and} \quad |x + 6| = -(x + 6)
\]
Substituting these in the inequality:
\[
-(3x - 1) < 2(-(x + 6))
\]
\[
-3x + 1 < -2x - 12
\]
\[
-x < -13 \Rightarrow x > 13
\]

Since \(x > 13\) does not overlap with \(x < -6\), this interval has no solutions.

#### Case 2: \(-6 \leq x < \frac{1}{3}\)
For this range, \(3x - 1 < 0\) and \(x + 6 \geq 0\), so:
\[
|3x - 1| = -(3x - 1) \quad \text{and} \quad |x + 6| = x + 6
\]
Substituting these:
\[
-(3x - 1) < 2(x + 6)
\]
\[
-3x + 1 < 2x + 12
\]
\[
-5x < 11 \Rightarrow x > -\frac{11}{5}
\]

Since \(-6 \leq x\), the solution in this interval is \(-\frac{11}{5} \leq x < \frac{1}{3}\).

#### Case 3: \(x \geq \frac{1}{3}\)
For this range, \(3x - 1 \geq 0\) and \(x + 6 \geq 0\), so:
\[
|3x - 1| = 3x - 1 \quad \text{and} \quad |x + 6| = x + 6
\]
Substituting these:
\[
3x - 1 < 2(x + 6)
\]
\[
3x - 1 < 2x + 12
\]
\[
x < 13
\]

Since \(x \geq \frac{1}{3}\), the solution in this interval is \(\frac{1}{3} \leq x < 13\).

### Step 4: Combine the Results
The solution set of the inequality is:

\[
x \in \left[-\frac{11}{5}, \frac{1}{3}\right) \cup \left[\frac{1}{3}, 13\right)
\]

### Summary
The values of \(x\) that satisfy the inequality \( |3x - 1| < 2|x + 6| \) are \(x \in \left[-\frac{11}{5}, 13\right)\).

---

Would you like to see a detailed explanation of each step, or do you have any questions about this solution?

### Related Questions:
1. How would the solution change if the inequality was \( |3x - 1| > 2|x + 6| \)?
2. What happens to the solution set if both sides of the inequality are squared?
3. How can we graph the functions \( y = |3x - 1| \) and \( y = 2|x + 6| \) to visualize the inequality?
4. How would you solve the inequality if a constant was added to both sides?
5. What if there were more than two absolute value expressions involved?

### Tip:
When solving inequalities involving absolute values, always consider the critical points where the expressions inside the absolute values change signs. These points will help to break down the problem into manageable cases.

Learn how to solve the inequality |3x - 1| < 2|x + 6| step-by-step with detailed explanations and examples. Understand the concept of absolute values and their impact on linear inequalities. Suitable for advanced high school students studying inequalities and absolute values.

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