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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: 
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