Nine and one-half less than four and one-half times a number is greater than 62.5. Which of the following represents the solution set of this problem?
A. $(16,+\infty)$
B. $(-16,+\infty)$
C. $(-\infty, 16)$
D. $(-\infty,-18)$
Answer (2)
The problem can be expressed as the inequality 62.5"> 4.5 x − 9.5 > 62.5 . By solving it, we find 16"> x > 16 , which represents the solution set in interval notation as ( 16 , + ∞ ) . Therefore, the correct answer is Option A: (16, +\infty).
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Translate the word problem into a linear inequality: 62.5"> 4.5 x − 9.5 > 62.5 .
Add 9.5 to both sides: 72"> 4.5 x > 72 .
Divide both sides by 4.5: 16"> x > 16 .
Express the solution set in interval notation: ( 16 , + ∞ ) .
Explanation
Translating the Word Problem Let's break down this word problem step by step to find the solution set. First, we need to translate the words into a mathematical inequality. The problem states: 'Nine and one-half less than four and one-half times a number is greater than 62.5'. Let's represent the unknown number with the variable x .
Forming the Inequality Four and one-half times the number can be written as 4.5 x . Nine and one-half less than this is 4.5 x − 9.5 . The problem tells us that this expression is greater than 62.5, so we can write the inequality as: 62.5"> 4.5 x − 9.5 > 62.5
Isolating the Variable Now, let's solve the inequality for x . To isolate x , we first add 9.5 to both sides of the inequality: 62.5 + 9.5"> 4.5 x − 9.5 + 9.5 > 62.5 + 9.5
72"> 4.5 x > 72
Solving for x Next, we divide both sides of the inequality by 4.5: \frac{72}{4.5}"> 4.5 4.5 x > 4.5 72
16"> x > 16
Expressing the Solution Set The solution to the inequality is 16"> x > 16 . This means that any number greater than 16 will satisfy the original condition. In interval notation, this is represented as ( 16 , + ∞ ) .
Final Answer Therefore, the solution set for this problem is ( 16 , + ∞ ) .
Examples
Imagine you're saving money for a new bicycle that costs $62.5. You already have $9.5 saved, and you plan to save $4.5 each week. This problem helps you determine how many weeks you need to save to have more than $62.5. By solving the inequality, you find out that you need to save for more than 16 weeks to reach your goal. Understanding inequalities helps you plan your savings and achieve your financial goals.
