First Question:
The term “70 pounds or less” means the inequality will take the form (current weight) ≤ 70 lb. The dog will lose 3 pounds per week, so after one week, he will weigh 97-3, or 94 pounds. After two weeks, he will weigh 97-(2*3) = 97-6 = 91 pounds. Another way of saying it is that the dog will weigh 97-(3*n), where n is the number of weeks since the vet set the goal. Thus, n=1 after one week, n=2 after two weeks, and so on. Thus, each week, n goes up by one and the dog weighs 97-(3*n). The inequality is then:
97-(3*n) ≤ 70 lb.
The problem is to solve for n at the time the dog weighs 70 lbs or less. Inequalities are no different than the equations you are used to which use the “=” in terms of operations that can be performed on them. So to get n, use the following procedure:
97-(3*n) ≤ 70 lb.
-(3*n) ≤ -27 Subtracting 97 from both sides of the inequality equation
(3*n) ≥ 27 Multiplying both sides of the inequality equation by -1. Notice that multiplying both sides by a negative number reverses the inequality. Example: -5 < 3, but 5 > -3.
n ≥ 27/3 Dividing both sides of the inequality equation by 3
n ≥ 9 Answer: It will take 9 weeks for the dog to reach the 70 lb goal. The equality part of the ≤ is satisfied.
FYI: Suppose the dogs weight had to be LESS than 70 lb. In that case, all the steps taken above would be the same, but end up as n>9. Thus, another week must pass (since the next higher value of n after 9 is 10), and it would take the dog ten weeks to get under 70 lb.
Second Question
The income from earbud sales is 6*n, where n is the number of earbuds sold. So the problem can be stated as
Income = $1500+(6*n)
To buy the new calculator, the income must equal or exceed the price of the calculator, $2100 (the term, “at least” implies greater than, or equal to). The problem then, is to find n, and the equation and solution is thus:
Income ≥ $2100
$1500+(6*n) ≥ $2100
6*n ≥ $2100 – 1500 Subtract $1500 from both sides of the inequality
6*n ≥ $600 Perform the subtraction
n ≥ $600/$6 Divide both sides of the inequality by six
n ≥ 100 Answer. The math club must sell 100 or more earbuds to buy the calculator
Third Question
First determine how much money the team has to buy volleyballs after buying the net:
Volleyball funds = $495-$195 = $300
Now if n is the number of volleyballs the team can buy at $17 each, the inequality is:
$17*n ≤ $300
n ≤ $300/$17 Divide both sides of the inequality by 17
n ≤ 17.65 volleyballs
Since there is no reason to buy approximately 2/3 of a volleyball, even if it were possible, they can still buy 17 volleyballs and have change left over.
