Q11 Solve the inequality Express your answer using interval

Q11. Solve the inequality. Express your answer using interval notation. Graph the solution set.

7 2x + 3 15
   a. (-6, -2)

   b. [-6, -2]

   c. [2, 6]

   d. (2, 6)


Q12. Multiply each side of the inequality 2 - 4x -2 by 5. What is the resulting inequality?
   a. 10 - 20x -10
   b. 10 - 20x -10
   c. 10 - 4x -10
   d. 10 - 4x -10

Q13. Translate the following sentence into a mathematical equation. Be sure to identify the meaning of all symbols.

The profit derived from the sale of x video cameras is $370 per unit less the sum of $2700 costs plus $160 per unit.
   a. If P is profit and x the units sold, then P = 370x - (2700 + 160x) or P = 210x - 2700.
   b. If P is profit and x the units sold, then P = 370x + 2700 - 160x or P = 210x + 2700.
   c. If P is profit and x the units sold, then P = 370x - (2700 - 160x) or P = 530x - 2700.
   d. If P is profit and x the units sold, then P = 370/x - (2700 + 160/x) or P = 210/x - 2700.

Q14. During a hurricane evacuation from the east coast of Georgia, a family traveled 210 miles west. For part of thetrip, they averaged 50 mph, but as the congestion got bad, they had to slow to 10 mph. If the total time of travel was 8 hours, how many miles did they drive at the reduced speed?
   a. 57.5 mi
   b. 42.5 mi
   c. 52.5 mi
   d. 47.5 mi

Q15. Find the real solutions of the equation x + x = 56.
   a. {8}
   b. {49}
   c. {7}
   d. {64}

Solution

11. 7 2x + 3 15 or, on adding -3 to both sides of the inequalities, 7-3 2x 15-3 or, 4 2x 12 so that, on dividing both sides of the inequalities by 2, we have 2 x 6. Thus, x [ 2,6].Option c) is the correct answer. The graph of the solution set is the segment of the real number line between ( and including) 2 and 6.

12. On multiplying each side of the inequality 2 - 4x -2 by 5, we get 10- 20x -10. Option a) is the correct answer.

13. The profit P derived from the sale of x video cameras is $370 per unit less the sum of $2700 costs plus $160 per unit or, P = 370x - (2700 + 160x) or P = 210x - 2700. x is the number of video cameras sold, $ 370 is the selling price of each video camera, 370x is the revenue, 2700 plus 160 per unit i.e. 2700+160x is the cost and other expenditure. Therefore, option a) is the correct answer.

14. Let the number of miles travelled at the reduced speed be x. Then the number of miles travelled at 50 mph is 210-x. Since time = distance/speed, the time taken in travelling 210-x miles and x miles are (210-x)/50 and x/10 respectively so that (210-x)/50 + x/10 = 8. On multiplying both the sides by 50, we get (210-x) +5x = 8*50 or, 210+4x = 400 or, 4x = 400-210 = 190. Hence x = 190/4 = 47.5 miles. Thus, option d) is the correct answer.

15. We have x+x=56 or,x-56 =-x. On squaring both the sides, we get (x-56)²=(-x)² or, x²-112x+3136 = x or, x²-113x+3136 = 0. Now, on using the quadratic formula, we get x = [-(-113)±{(-113)² -4*1*3136}]/2 =[113±(12769-12544)]/2 = (113±225)/2 = (113±15)/2. Thus either x = (113+15)/2 = 64 or, x =                     (113-15)/2 = 49. Option b) and d) are the correct answers.

Note:

For a quadratic equation, either both the solutions are real or both are complex as complex roots of a polynomial occur in conjugate pairs.