Applications of Linear and Quadratic Equations Solving Verbal Problems Read the question carefully. Choose a letter to represent the unknown quantity. Use facts and relationships from the problem to create equation(s) involving the letter. Solve the equation. Check the solution to see if it answers the question. Interpret and then explain the meaning of the solution found.

Basic Relationships profit = total revenue - total cost total revenue = (price per unit)(number of units sold) = p  q total cost = variable cost + fixed cost

Example 1 A person invested $20 000: part at an interest rate of 6% annually and the remainder at 7% annually. The total interest at the end of 1 year was equivalent to an annual 6 ¾ % rate on the entire $20 000. How much was invested at each rate? Simple Interest = Principal.rate.time SI = P×r×t Let the value invested at 6% be x. Then the amount invested at 7% is (20 000 - x).

Looking at the interest earned in 1 year: $5000 invested at 6 % and $15000 invested at 7%.

Example 2 The cost of a product to a retailer is $3.40. If the retailer wishes to make a profit of 20% on the selling price, at what price should the product be sold? Let the selling price be x.

Cost = $3.40 Profit = 20% of selling price = 0.2x The product should be sold for $4.25.

Example 3 A college dormitory houses 210 students. This fall (autumn), rooms are available for 76 freshman (first-years). On the average, 95% of those freshman who request room applications actually reserve a room. How many room applications should the college send out if it wants to receive 76 reservations? Let the number of applications be x.

The college should forward 80 applications for rooms.

Example 4 A compensating balance refers to that practice wherein a bank requires a borrower to maintain on deposit a certain portion of a loan during the term of the loan. For example, if a firm makes a $100000 loan which requires a compensating balance of 20%, it would have to leave $20000 on deposit and would have the use of $80000.

To meet the expenses of retooling, the Victor Manufacturing Company must borrow $95000. The Third National Bank, with whom they have had no prior association, requires a compensating balance of 15%. To the nearest thousand dollars, what must be the amount of the loan to obtain the needed funds? Let the amount to be borrowed be x.

If 15% is the compensating balance, then $95000 must represent 85% of the loan. The company should borrow $112 000.

Inequalities An inequality is a statement about the order of two numbers. That is, that one number is less than another. Symbols to show this order are:

Examples 7 ) 2 6 )

Solving Inequalities Care must be taken to maintain the sense of the inequality. The allowable operations are the same as those for equations with two exceptions. 1. When you multiply both sides of the inequality by a negative number the sense is reversed. 2. When finding the reciprocal of each side, the sense is reversed.

Example

Example

Example t (

Example y )

Example A company produces alarm clocks. During the regular work week, the labour cost for producing one clock is $2.00. However, if a clock is produced in overtime the labour cost is $3.00. Management has decided to spend no more than a total of $25000 per week for labour. The company must produce 11000 clocks this week. What is the minimum number of clocks that must be produced during the regular work week?

Let the number of clocks that must be produced in a regular week be x. Then the number produced in overtime is 11000 - x. The minimum number to be produced is 8000.

Example The current ratio of Precision Machine Products is 3.8. If their current assets are $570 000, what are their current liabilities? To raise additional funds, what is the maximum amount they can borrow on a short-term basis if they want their current ratio to be no less than 2.6? (See Example 3 in Section 1.3 of the textbook for an explanation of current ratio).

Current ratio = assets liabilities Current liabilities are $150 000.

The company can borrow at most $112 500 on a short term basis.

Absolute Value The absolute value of a number can be thought of as its distance from zero. eg 4 and -4 are both 4 units from the origin. They both have an absolute value of 4.

Absolute Value Note: 1. The absolute value is always positive. 2. 3. 4.

Example -8 8 x - 5 x - 5

Equations and Inequalities Involving Absolute Value Example -2 2 ( ) x + 7

Example -4 4