SOLUTION: Two times the difference between a number and eleven is less than or equal to seven times the sum of the number and four. Find the smallest number that will satisfy the inequality.

Question 1014301: Two times the difference between a number and eleven is less than or equal to seven times the sum of the number and four. Find the smallest number that will satisfy the inequality.
what i tried..
2(n-11) less than or equal to sign 7(n+4)
simplified to 50 less than or equal to 5n
divided 5 by each side ended up with 10
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you're on the right track.
however, the result is n >= -10.
here's why.

start with 2 * (n-11) <= 7 * (n+4)
distribute the multiplication to get 2n - 22 <= 7n + 28
subtract 7n from both sides of the equation and add 22 to both sides of the equation to get 2n - 7n <= 28 + 22
combine like terms to get -5n <= 50

here's where the inequality gets flipped.

divide both sides of the inequality by -5 to get n >= 50/-5
simplify to get n >= -10

when you multiply or divide both sides of an inequality by a negative number, the inequality gets reversed.

you have a > b.
multiply both sides of this inequality by -1 and you get -a < -b.
in numbers, you would get the following:
9 > 5
multiply both sides by -1 and you get -9 < -5.

the equality part remains the same.
7 = 7
multiply both sides of this equation by -1 and you get -7 = -7
they are still equal to each other, only with different signs.

in your inequality, you had -5n <= 50
division is a form of multiplication and the same rules apply.
dividing by 5 is the same as multiplying by 1/5.
when you divided by -5, the inequality changes.
you wind up with n >= -10.

i confirmed the solution is correct by replacing n with -11, -10, -9.
the results are what i expected.
when n was -11, 2*(n-11) <= 7*(n+4) was false (not smaller or equal).
when n was -10, 2*(n-11) <= 7*(n+4) was true (equal).
when n was -9, 2*(n-11) <= 7*(n+4) was true (smaller)

Answer by MathTherapy(10555) (Show Source): You can put this solution on YOUR website!

Two times the difference between a number and eleven is less than or equal to seven times the sum of the number and four. Find the smallest number that will satisfy the inequality.
what i tried..
2(n-11) less than or equal to sign 7(n+4)
simplified to 50 less than or equal to 5n
divided 5 by each side ended up with 10

Number's ≥ - 10, so SMALLEST NUMBER/INTEGER is:
RELATED QUESTIONS
1.the sum of a number and fourteen is less or equal to three times the number 2.negative (answered by mananth)
Four times a number is less than or equal to the sum of three times the number and... (answered by ikleyn)
Eleven more than five times a number is equal to the difference between 227 and four... (answered by Maths68)
Translate each of the following word expressions into an Algebraic Expression 1.Twelve (answered by rfer)
eleven times the difference of a number and seven (answered by ewatrrr)
wright these is equations 1: seventeen is four times the sum of a number and two... (answered by solver91311)
Four times a number is less than or equal to the sum of 3 times the number and-2 (answered by stanbon)
translate :the difference between seven times a number and eleven is... (answered by edjones)
The sum of two numbers is 18. The difference between four times the smaller number and... (answered by checkley77)