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CALCULUS AB WORKSHEET THREE WARM-UP AND REVIEW “Inequalities” [1.] Explain, define or illustrate the following terms or phrases: intersection, algebraic notation, set notation, interval notation, union, inequality, simple inequality, compound inequality, sign plot, rational inequality, comparison to zero, and quadratic inequality. [2.] Explain the process of solving a. Simple Inequality b. Compound Inequality c. Complex Inequality SOLVE EACH INEQUALITY BY USING THE CIS METHOD - SHOW ALL OF YOUR WORK – REPRESENT SOLUTIONS IN INTERVAL NOTATION – VERIFY SOLUTIONS USING A GRAPHING CALCULATOR [3.] 4x 11 x 1 3x 7 [4.] 9x 8x 2 3x [5.] 3x 5 x2 [6.] x 1 3 x [7.] x2 2 x 1 [8.] 4 2 3 7 x x [9.] 3x 8 2 2x 3 3 [10.] x 2 2x 15 0 [11.] 4 x 2 3x [12.] x 10 3x 2 [13.] x3 0 x2 4 [14.] 4 1 2x 1 x 1 [15.] [17.] x2 9 2 x 3x 2 0 [16.] 3x 1 2 x x6 1 A firm can sell at a price of $100.00 per unit all the “Disraeli Gears” it produces. If x units are produced each day, the number of dollars in the total cost of each day’s production is given by the expression: x 2 20x 700 Page 1 of 5 Revised 07/24/2004 gfmartin CALCULUS AB WORKSHEET THREE WARM UP AND REVIEW – INEQUALITIES Determine how many units should be produced each day so that the firm is guaranteed a profit. Page 2 of 5 Revised 07/24/2004 gfmartin CALCULUS AB WORKSHEET THREE WARM UP AND REVIEW – INEQUALITIES [18.] A rectangular plot of ground is to be enclosed by a fence and then divided down the middle by another fence. The fence down the middle costs $3.00 per running foot and the perimeter fence costs $6.00 per running foot. The area of the plot is to be exactly 1800 square feet and the cost of the fence is not to exceed $2310.00. Determine the restrictions on the dimensions of the plot. SOLUTIONS [1.] Intersection refers to what two solutions (sets) have in common. Algebraic, Set and Interval Notation are different ways of writing the solution to an equation or inequality. expression compared to zero. Then determine where the expression is positive or negative by using a sign plot. The sign plot can then be used to determine the solution. [3.] Union is the combination of two solutions (sets). x (2, ) [4.] 9x 8x 2 U 8x 2 3x x 172 U x 25 An inequality is any comparison statement using the symbols <. ≤, > or ≥. x Isolating the x term on one side by addition and/or multiplication can solve a simple inequality. The solution will be the intersection of the two individual solutions. Since the individual solutions have no elements in common, the final solution is the null set. A compound inequality is any quality containing two comparison symbols. The inequality must be writing as two separate inequalities and solved individually. [5.] x , 5 [6.] x1 3 0 x x 1 3x 0 x 4x 1 0 x A sign plot is a way of illustrating where a variable expression is positive or negative by first determining where it is equal to zero. Complex inequalities (rational or quadratic) are best solved by comparison to zero and then using a sign plot to obtain a solution. [2.] A simple inequality is solved by algebraically isolating the variable on the left or right side of the inequality symbol by addition and/or multiplication. Once the variable is isolated the result is the algebraic solution to the inequality. A compound inequality of the form A B C can be solved by splitting the inequality into two distinct inequalities and solving each separately by then taking the intersection of the two separate solutions. 2, Create a sign plot to determine the intervals the rational expression is zero and positive. x , 14 [7.] (0, ) x 1 , 4 [8.] A complex inequality is best solved by comparison to zero. Rewrite the given inequality so that the result is a variable Page 3 of 5 Revised 07/24/2004 gfmartin CALCULUS WORKSHEET THREE WARM UP AND REVIEW – INEQUALITIES 4 2 3 7 x x 4 2 3 7 0 x x 2 40 x 2 4x 0 x Create a sign plot to determine the intervals the rational expression is zero and positive. x , 12 [9.] Create a sign plot to determine the intervals the rational expression is negative. x , 5 2 [15.] x 3,1 [16.] (0,) [10.] x 2 2x 15 0 (x 5)(x 3) 0 Create a sign plot to determine the intervals where the product of the factors is zero and positive. [11.] x ( , 4] [3,) x (, 2) [12.] x 10 3x 2 3x x 10 0 (3x 5)(x 2) 0 2 Create a sign plot to determine the intervals where the product of the factors is positive. [13.] x 2,2 Create a sign plot to determine the intervals the rational expression is zero and negative. [1, ) x , 2 2, 3 3x 1 1 x2 x 6 3x 1 1 0 x2 x 6 3x 1 1x 2 x 6 0 x2 x 6 x 2 4x 5 0 x2 x 6 1 x 2 4x 5 0 x2 x 6 1(x 5)(x 1) 0 (x 3)(x 2) x 23 ,6 x (, 5] –1, 12 [1, 3) [5,) [17.] Producing between 10 and 70 units daily will result in a profit. [18.] Let x be the horizontal length of the fence and y vertical length. Therefore the total amount (T) of fencing can be defined by the equation T x x y y y 53 , The total cost (C) of the fencing can be defined by the equation 3, C 6x 6x 6y 6y 3y 12x 15y [14.] 4 1 2x 1 x 1 4 1 0 2x 1 x 1 4(x 1) 1(2x 1) 0 (2x 1)(x 1) Since we do not want the cost of the fence to exceed $2310 we can define the cost inequality in the following manner C 2310 12x 15y 2310 2x 5 0 (2x 1)(x 1) Note that we have two variables. So we must eliminate either x or y. This is a common Page 4 of 5 Revised 07/24/2004 gfmartin CALCULUS WORKSHEET THREE WARM UP AND REVIEW – INEQUALITIES type of situation. You must be given an initial condition. In this case the initial condition is the restriction on the area. The area (A) can be defined as follows: A xy 1800 Solve the area equation for x or y and substitute into the cost inequality. x 1800 y 1800 12 15y 2310 y Since you know that y is always a positive value you can multiply each side by y without affecting the final outcome. In other words multiplying an inequality by a positive value does not effect the direction of the inequality. 12 1800 15y 2310 y 15y 2 2310y 21600 0 15 y 144 y 10 0 Create a sign plot to find where the variable expression is negative. Since y is a positive value you need not consider in values of y that are negative. Sign plot should reveal that if y is a value in the interval [10,144] you have satisfied the requirements of the problem. To obtain the x restrictions, substitute the endpoints of 10 and 144 in for y into the initial condition and obtain the interval [12.5,180] Therefore vertical length that is between 10 feet and 144 feet (inclusive) paired with the appropriate horizontal length of 12.5 feet to 180 feet (inclusive) will result in a rectangular plot of 1800 square feet and not exceed the price of fencing of $2310. Page 5 of 5 Revised 07/24/2004 gfmartin