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In Mathematics / College | 2025-07-08

Select the correct answer.
If [tex]$5(4-x)\ \textless \ y+12$[/tex] and [tex]$y+12\ \textless \ 3 x+1$[/tex], then which statement is true?
[tex]$3 x+1-5(4-x)=y+12$[/tex]
[tex]$3 x+1\ \textless \ 5(4+x)$[/tex]
[tex]$5(4-x)\ \textless \ 3 x+1$[/tex]
[tex]$5(4-x)+3 x+1=y+12$[/tex]

Asked by jordanmariexox

Answer (2)

Combine the given inequalities to get 5 ( 4 − x ) < y + 12 < 3 x + 1 .
Deduce that 5 ( 4 − x ) < 3 x + 1 .
Compare the result with the given options.
The correct statement is 5 ( 4 − x ) < 3 x + 1 ​ .

Explanation

Understanding the Problem We are given two inequalities: 5 ( 4 − x ) < y + 12 and y + 12 < 3 x + 1 . We want to find the correct statement based on these inequalities.

Combining the Inequalities Since 5 ( 4 − x ) < y + 12 and y + 12 < 3 x + 1 , we can combine these inequalities into a single inequality: 5 ( 4 − x ) < y + 12 < 3 x + 1 . This tells us that 5 ( 4 − x ) is less than 3 x + 1 .

Finding the Correct Statement Therefore, we have 5 ( 4 − x ) < 3 x + 1 . Now we compare this with the given options.

Analyzing the Options The first option is 3 x + 1 − 5 ( 4 − x ) = y + 12 . This is not necessarily true, as we only know that 5 ( 4 − x ) < y + 12 < 3 x + 1 .
The second option is 3 x + 1 < 5 ( 4 + x ) . Let's expand this: 3 x + 1 < 20 + 5 x . This is not directly derived from the given inequalities. The third option is 5 ( 4 − x ) < 3 x + 1 . This is exactly what we derived from the given inequalities. The fourth option is 5 ( 4 − x ) + 3 x + 1 = y + 12 . This is not necessarily true, as we only know that 5 ( 4 − x ) < y + 12 < 3 x + 1 .

Conclusion Thus, the correct statement is 5 ( 4 − x ) < 3 x + 1 .


Examples
In real life, inequalities like these can be used to determine constraints in resource allocation. For example, if you have a limited budget and need to allocate funds between different projects, these inequalities can help you determine the feasible range of spending for each project while staying within your budget and meeting certain requirements. Understanding how to manipulate and combine inequalities is crucial for making informed decisions in various fields, from finance to engineering.

Answered by GinnyAnswer | 2025-07-08

After analyzing the given inequalities, the statement 5 ( 4 − x ) < 3 x + 1 is derived and is true. This conclusion is reached by combining and interpreting the inequalities provided. Therefore, the correct option is 5 ( 4 − x ) < 3 x + 1 .
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Answered by Anonymous | 2025-07-09

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