We will start by solving the inequality. Then, we will find the probability that a point chosen at random from AK satisfies the inequality.
Solving the Inequality
The given compound inequality can be written as an AND inequality.
2≤ 4x ≤ 3
⇕
2≤ 4x AND 4x≤ 3
We can solve these inequalities one at a time. Let's start by solving 2≤ 4x.
Now, let's solve 4x≤ 3.
4x≤ 3 ⇔ x≤3/4
We found that x is greater than or equal to 12 andless than or equal to 34. We can recombine these solution sets to express this as a single compound inequality.
x≥ 1/2 AND x≤ 3/4
⇕
1/2≤ x≤ 3/4
Finding the Probability
We can use geometric models to solve certain types of probability problems. In geometric probability, points on a segment or in a region of a plane represent outcomes. The geometric probability of an event is a ratio that involves geometric measures such as length or area. Consider the given diagram.
We are told that a point on AK is chosen at random, and want to find the probability that the point satisfies the inequality 12≤ x ≤ 34. Let A'B' be the segment defined by this inequality.
The probability that the point is on A'B' is the ratio of the length of A'B' to the length of AK.
P(The point is onA'B')=A'B'/AK
Looking at the given number line, we see that AK= 10 and A'B'= 0.25
We can substitute these values in the above formula to find the probability that the point lies on A'B'. This is the same as finding the probability that the point satisfies the given inequality.
