The triangle inequality says the two smallest sides have to add up to strictly more than the biggest. So when 7 and 12 are the smaller sides the biggest can't be 19 or larger. When 12 is the largest side the smallest side can't be 5 or smaller. So we check those in the open interval (5,19):
According to the triangle inequality theory the sum of 2 sides of a triangle need to be grater than the 3rd side.
If we sum 7 + 12 we obtain a result of 19. Since nineteen is the answer, we select all the options that summed with either 12 or 7 are lower than 19 but greater than 5 (because if you sum up 5 with the numbers, the answers won't match according to the theory):
Let's define variables: x = Rate in $ / hr of the mechanic working 15hr y = Rate in $ / hr of the mechanic working 5hr We have the following system of equations: x + y = 130 15x + 5y = 1400 Solving the system we have: x = $ 75 / hr y = $ 55 / hr Answer: the rate charged per hour by mechanics was: x = $ 75 / hr y = $ 55 / hr
