To solve this problem, we need to determine the probability that a randomly chosen number from the set [tex]\(\{-3, -2, -1, 0, 1, 2\}\)[/tex] satisfies the inequality [tex]\(4 - 3x < 6\)[/tex].
1. Solve the Inequality:
[tex]\[
4 - 3x < 6
\][/tex]
Subtract 4 from both sides:
[tex]\[
-3x < 2
\][/tex]
Divide both sides by -3 (and reverse the inequality since we are dividing by a negative number):
[tex]\[
x > -\frac{2}{3}
\][/tex]
2. Identify the Valid Values:
We next check which numbers in the set [tex]\(\{-3, -2, -1, 0, 1, 2\}\)[/tex] are greater than [tex]\(-\frac{2}{3}\)[/tex]:
- -3: Not greater than [tex]\(-\frac{2}{3}\)[/tex]
- -2: Not greater than [tex]\(-\frac{2}{3}\)[/tex]
- -1: Not greater than [tex]\(-\frac{2}{3}\)[/tex]
- 0: Is greater than [tex]\(-\frac{2}{3}\)[/tex]
- 1: Is greater than [tex]\(-\frac{2}{3}\)[/tex]
- 2: Is greater than [tex]\(-\frac{2}{3}\)[/tex]
Therefore, the valid values from the set are [tex]\(\{0, 1, 2\}\)[/tex].
3. Calculate the Probability:
There are 3 numbers that meet the condition [tex]\(x > -\frac{2}{3}\)[/tex], and the total number of elements in the set is 6.
The probability [tex]\(P\)[/tex] is given by the ratio of the number of valid outcomes to the total number of outcomes:
[tex]\[
P = \frac{\text{Number of valid values}}{\text{Total number of values}} = \frac{3}{6} = \frac{1}{2}
\][/tex]
Therefore, the probability that [tex]\(x\)[/tex] will satisfy the inequality [tex]\(4 - 3x < 6\)[/tex] when picked at random from the set [tex]\(\{-3, -2, -1, 0, 1, 2\}\)[/tex] is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
