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If \(m^2 < 225\) and \(n - m = -10\), what is the difference between the smallest possible integer value of \(3m + 2n\) and the greatest possible integer value of \(3m + 2n\)?

If \(m^2 < 225\) and \(n - m = -10\), what is the difference between the smallest possible integer value of \(3m + 2n\) and the greatest possible integer value of \(3m + 2n\)?

A. -190 B. -188 C. -150 D. -148 E. -40

This question is about algebraic manipulations with inequalities.

From \(n - m = -10\) it follows that \(n=m-10\). Thus, \(3m + 2n=3m+2(m-10)=5m-20\). So, we need to find the difference between the smallest possible integer value of \(5m-20\) and the greatest possible integer value of \(5m-20\).

Now, lets' work on \(m^2 < 225\):

Take the square root from both sides: \(|m| < 15\);

Get rid of the modulus sign: \(-15 < m < 15\);

Multiply all three parts by 5: \(-75 < 5m < 75\);

Subtract 20 from all three parts: \(-95 < 5m -20 < 55\);

From \(-95 < 5m -20 < 55\) it follows that the smallest possible integer value of \(5m-20\) is -94 and the greatest possible integer value of \(5m-20\) is 54.

For the below solution is it also possible to directly put the least possible value of m(-14) and the greatest possible value of m(14) in the equation( 5m-20)?

Regards, Mahuya

If \(m^2 < 225\) and \(n - m = -10\), what is the difference between the smallest possible integer value of \(3m + 2n\) and the greatest possible integer value of \(3m + 2n\)?

A. -190 B. -188 C. -150 D. -148 E. -40

This question is about algebraic manipulations with inequalities.

From \(n - m = -10\) it follows that \(n=m-10\). Thus, \(3m + 2n=3m+2(m-10)=5m-20\). So, we need to find the difference between the smallest possible integer value of \(5m-20\) and the greatest possible integer value of \(5m-20\).

Now, lets' work on \(m^2 < 225\):

Take the square root from both sides: \(|m| < 15\);

Get rid of the modulus sign: \(-15 < m < 15\);

Multiply all three parts by 5: \(-75 < 5m < 75\);

Subtract 20 from all three parts: \(-95 < 5m -20 < 55\);

From \(-95 < 5m -20 < 55\) it follows that the smallest possible integer value of \(5m-20\) is -94 and the greatest possible integer value of \(5m-20\) is 54.

For the below solution is it also possible to directly put the least possible value of m(-14) and the greatest possible value of m(14) in the equation( 5m-20)?

Regards, Mahuya

We are NOT told that m is an integer, hence from -15<m<15 saying that the minimum value of m is -14 and the maximum value of m is 14 is wrong.
_________________

For the below solution is it also possible to directly put the least possible value of m(-14) and the greatest possible value of m(14) in the equation( 5m-20)?

Regards, Mahuya

We are NOT told that m is an integer, hence from -15<m<15 saying that the minimum value of m is -14 and the maximum value of m is 14 is wrong.

That is exactly where I got stuck and ended up guessing the answer to be the closest number I got with max = 14 min = -14. Gotta read the question more carefully!

I am a little bit confused as for the meaning of "the difference between x and y". In this question, the wording has not effect on the solution since all the answers are negative. However, it may cause trouble in future problems. That being said, does "the difference between x and y" get translated to "x-y" or to "y-x"?

Thanks for your help!

-- Need Kudos
_________________

If you find this post hepful, please press +1 Kudos

I have a question on this. The mentioned answer will work only when m =+-15, which is not possible as m2 has to be lass that 225. Also this question does mention that 3m+2n should be greatest and minimum INTEGER. Should it make any difference in the answer?

I have a question on this. The mentioned answer will work only when m =+-15, which is not possible as m2 has to be lass that 225. Also this question does mention that 3m+2n should be greatest and minimum INTEGER. Should it make any difference in the answer?

Given information : Square(m) < 225 => -15<m<15 --- (1) & n-m=10 ---(2)

Manipulation starts : We will fit n in the inequality using (1) & (2) as that is the demand of the question. Replace m in (1) with n-10 from (2), new equation will be -25<n<5 --- (3)

Now, simply make (1) & (3) in the form of equation as asked in the question i.e. 3m+2n -45<3m<45 (Multiplying (1) by 3) -50<2n<10 (Multiplying (3) by 2)

Add above two equations , we will get -95<3m+2n<55

Remember integer criteria is applicable on 3m+2n & not on m & n separately.

From above equation, Min integer = -94 & Max integer is 54 Difference = -94-54 = -148

What if we take m = -14 , n = -24 and m = 14 , n = 4 and try to find the answer?

Have you tried that? Did you get the correct answer?

We are NOT told that m is an integer, hence from -15<m<15 saying that the minimum value of m is -14 and the maximum value of m is 14 is wrong.
_________________