inequalities and word problems

Question

A tea merchant mixes some black tea costing $42 per pound with 3 lb of Earl Grey tea costing $50 per pound. How many pounds of the black tea should be used if the merchant wants a blend that costs between $45 and $46 per pound?

Answer 1

Let x be the variable that represents in pounds, the amount of black tea the merchant will add; therefore,   45≤ [42x + 50(3)]/(x +3) ≤ 46

 

Now solve the compound inequality in two steps:

 

the left side: how many pounds of the black tea so that the mixture will cost $45 per pound-

• 45= [42x + 50(3)]/(x+3)

         45x + 135 = 42x + 150

          3x  = 15

             x = 5

the right side: how many pounds of the black tea so that the mixture will cost $46 per pound-

• 46= [42x + 50(3)]/(x+3)

          46x + 138 = 42x + 150
            4x = 12
              x = 3

 

However, the question is to determine the amount in pounds of the black tea so that the mixture will cost between $45-$46 per pound.

 

The answer is, between 3 and 5 pounds of the black tea OR 3 ≤ x ≤ 5.