Solve the compound inequality $5 \leq x+2<11$.
A) $9
B) $3 \leq x<9$
C) $3
D) $7 \leq x<13$
Answer (1)
Subtract 2 from all parts of the inequality: 5 − 2 ≤ x + 2 − 2 < 11 − 2 .
Simplify the inequality: 3 ≤ x < 9 .
The solution is 3 ≤ x < 9 .
The correct answer is B) 3 ≤ x < 9 .
Explanation
Understanding the Problem We are given the compound inequality 5 ≤ x + 2 < 11 . Our goal is to isolate x in the middle to find the range of values that satisfy the inequality.
Isolating x To isolate x , we need to subtract 2 from all parts of the inequality. This gives us: 5 − 2 ≤ x + 2 − 2 < 11 − 2
Simplifying the Inequality Now, we simplify each part of the inequality: 3 ≤ x < 9
Finding the Solution This means that x is greater than or equal to 3, and strictly less than 9. In other words, x can be 3, but it cannot be 9. Looking at the answer choices, we see that option B matches this result.
Examples
Imagine you're planning a surprise birthday party and need to keep the number of guests within a certain range. You want at least 5 guests but no more than 10. If you've already invited 2 people, this inequality helps you determine how many more people you can invite. By solving 5 ≤ x + 2 < 11 , you find that you can invite between 3 and 8 more people to stay within your desired guest range. This kind of problem appears in resource allocation, project management, and even in everyday decision-making where constraints apply.
