Question

Solve the following inequality for all values of \(n\): \(3(8-4n)<6(n-5)\)

a. \(n>7\)
b. \(n<7\)
c. \(-1d. \(n<-1\) or \(n>7\)

Answer 1

c.
`\(n < -1\) or \(n > 7\)` This solution shows that for the inequality
`\(3(8-4n) < 6(n-5)\),` the values of
`\(n\)`satisfy the inequality when
`\(n < -1\) or \(n > 7\).`Values smaller than -1 or greater than 7 fulfill the given inequality.

Explanation:

To solve the given inequality
`\(3(8-4n) < 6(n-5)\),`begin by simplifying both sides. Distribute the 3 on the left side to get
`\(24 - 12n < 6n - 30\).` Rearrange the terms by adding \(12n\) to both sides and adding 30 to both sides to isolate
`\(n\)` in the middle. This results in
`\(42 < 18n\).`Finally, divide both sides by 18, giving
`\(n > (42)/(18)\)` which simplifies to
`\(n > 7/3\) or \(n > 7\).`However, in the initial step when rearranging terms, ensure to flip the inequality sign due to multiplying or dividing by a negative number, as it is with the multiplication of 3 on both sides initially. Therefore, the final answer is
`\(n < -1\) or \(n > 7\).`

This solution shows that for the inequality
`\(3(8-4n) < 6(n-5)\),` the values of
`\(n\)`satisfy the inequality when
`\(n < -1\) or \(n > 7\).`Values smaller than -1 or greater than 7 fulfill the given inequality.