Question

The solution set of the inequality \( |3x| \geq |6 - 3x| \) is: 

• A. (−∞, 1]
• B. [1, ∞)
• C. (−∞, 1) ∪ (1,∞)
• D. (−∞, −1) ∪ (−1,∞)

Hint:

When solving inequalities involving absolute values, it’s important to break them into different cases based on the possible signs of the expressions inside the absolute values. Each case leads to a different inequality that you can solve. If you end up with a contradiction (like \( 0 \leq -6 \)), that case provides no valid solutions. Once all cases are considered, the union of their solutions will give you the final answer.

Answer 1

The Correct Option is B

To solve the inequality \( |3x| \geq |6 - 3x| \), we need to consider the properties of absolute values. The absolute value of a number \( a \), \( |a| \), can be defined as: 

• \(|a| = a\) if \( a \geq 0 \)
• \(|a| = -a\) if \( a < 0 \)

Let's consider the expression \( |3x| \) and \( |6 - 3x| \) separately:

• \(|3x| = 3x\) if \( x \geq 0 \) and \(|3x| = -3x\) if \( x < 0 \)
• \(|6 - 3x| = 6 - 3x\) if \( 6 - 3x \geq 0 \) and \(|6 - 3x| = -(6 - 3x)\) if \( 6 - 3x < 0 \)

Next, we solve for the critical points that determine intervals for \( x \). Set \( 6 - 3x = 0 \), solving gives \( x = 2 \).

We analyze the inequality in three intervals determined by \( x = 0 \), \( x = 2 \):

• Interval 1: \( x < 0 \)
• Interval 2: \( 0 \leq x < 2 \)
• Interval 3: \( x \geq 2 \)

Evaluate the inequality in each interval:

• For \( x < 0 \): We have \(-3x \geq -(6 - 3x)\). This simplifies to \(-3x \geq -6 + 3x\) or \(0 \geq -6 + 6x\). Further simplifying gives \(6x \geq 6\) or \(x \geq 1\). But this is not possible for \( x < 0 \).
• For \( 0 \leq x < 2 \): We have \(3x \geq 6 - 3x\). Simplifying gives \(6x \geq 6\) or \(x \geq 1\). This means \(x \in [1,2)\).
• For \( x \geq 2 \): We have \(3x \geq 3x - 6\). Simplifying gives \(6 \geq 0\), which is always true for \( x \geq 2\).

Combine the solutions from the intervals. The overall solution set is \( x \in [1, \infty) \).

Thus, the solution set of the inequality is \([1, \infty)\).

Answer 2

The inequality \( |3x| \geq |6 - 3x| \) involves absolute values, so we need to split it into cases based on the definitions of absolute value:

Step 1: Case 1 - \( 3x \geq 6 - 3x \):

In this case, we assume that the expressions inside the absolute values are positive. Solving this inequality: \[ 3x + 3x \geq 6 \implies 6x \geq 6 \implies x \geq 1. \] Therefore, the solution for this case is \( x \geq 1 \).

Step 2: Case 2 - \( 3x \leq -(6 - 3x) \):

In this case, we assume that the expression inside the absolute value on the right side is negative. Solving this inequality: \[ 3x \leq -6 + 3x \implies 0 \leq -6, \] which is a contradiction (since 0 is not less than or equal to -6). Thus, this case has no solution.

Step 3: Conclusion:

From Case 1, we get \( x \geq 1 \), and Case 2 does not contribute any solutions. Therefore, the solution to the inequality is: \[ x \geq 1 \quad \text{or} \quad [1, \infty). \]