If x represents a number picked at random from the set `{-3, -2, -1, 0, 1, 2}`, what is the probability that x will satisfy the inequality `4-3xlt6`?

Options

A. `(1)/(6)`
B. `(1)/(3)`
C. `(1)/(2)`
D. `(2)/(3)`

Answer

To solve the problem, we need to determine the probability that a randomly selected number \( x \) from the set \(\{-3, -2, -1, 0, 1, 2\}\) satisfies the inequality \( 4 - 3x < 6 \).

### Step-by-Step Solution:

1. **Write down the inequality**:
\[ 4 - 3x < 6 \]

2. **Subtract 4 from both sides**:
\[ 4 - 3x - 4 < 6 - 4 \]
This simplifies to:
\[ -3x < 2 \]

3. **Divide both sides by -3**:
When dividing by a negative number, remember to reverse the inequality sign:
\[ x > \frac{2}{-3} \]
This simplifies to:
\[ x > -\frac{2}{3} \]

4. **Identify the values in the set that satisfy the inequality**:
The set is \(\{-3, -2, -1, 0, 1, 2\}\). We need to find which of these values are greater than \(-\frac{2}{3}\):
- \(-3\) is not greater than \(-\frac{2}{3}\)
- \(-2\) is not greater than \(-\frac{2}{3}\)
- \(-1\) is not greater than \(-\frac{2}{3}\)
- \(0\) is greater than \(-\frac{2}{3}\)
- \(1\) is greater than \(-\frac{2}{3}\)
- \(2\) is greater than \(-\frac{2}{3}\)

Therefore, the values that satisfy the inequality are \(0, 1, 2\).

5. **Count the number of favorable outcomes**:
There are 3 values that satisfy the inequality: \(0, 1, 2\).

6. **Count the total number of outcomes**:
The total number of elements in the set is 6.

7. **Calculate the probability**:
The probability \( P \) that \( x \) satisfies the inequality is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{6} = \frac{1}{2} \]

### Final Answer:
The probability that \( x \) will satisfy the inequality \( 4 - 3x < 6 \) is \(\frac{1}{2}\).

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