A tea merchant mixes some black tea costing $42 per pound with 3 lb of Earl Grey tea costing $52 per pound. How many pounds of the black tea should be used if the merchant wants a blend that costs between $45 and $47 per pound?

The amount in pounds of the black tea that should be used if the merchant wants a blend that costs between $45 and $47 per pound should be between 3pounds and 7 pounds. That is,

3 < x < 7.

Step-by-step explanation:

Let the amount of black tea mixed be x pounds

Total pounds of tea mixed = (x + 3) pounds

Amount spent on black tea = $42x

Amount spent on Earl Grey tea = 52 * 3 = $156

Total Amount spent on the tea = $ (156 + 42x)

Average amount spent on tea per pound = (156 + 42x) / (3 + x)

And this average amount is supposed to be between $45 and $47

Starting with the lower limit, $45,

[ (156 + 42x) / (3 + x) ] > 45

Since x is never negative,

156 + 42x > 45 (x + 3)

45x + 135 < 156 + 42x

45x - 42x < 156 - 135

3x < 21

x < 7

Then taking the upper limit,

[ (156 + 42x) / (3 + x) ] < 47

Since x is never negative,

156 + 42x < 47 (x + 3)

47x + 141 > 156 + 42x

47x - 42x > 156 - 141

5x > 15

x > 3

This means that x can be between 3 and 7,

The right range is 3 < x < 7 for the average to fall between $45 and $47.

3 ≤ x ≤ 7.

Step-by-step explanation:

Let X be the variable that represents in pounds, the amount of black tea the merchant will add; therefore,

45 ≤ [42x + (52 * 3) ] / (x + 3) ≤ 47

Solving;

The left side of the equation:

45 = [42x + (52 * 3) ] / (x+3)

45x + 138 = 42x + 156

3x = 21

x = 7

The right side of the equation:

47 = [42x + (52 * 3) ] / (x+3)

47x + 141 = 42x + 156

5x = 15

x = 3

The answer is 3 ≤ x ≤ 7