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distance traveled + 2 ≥ 10 To find the distance traveled, a modification of the speed formula will be used. r = d/t In this formula, r is the speed or rate, d is the distance traveled, and t is the time it took to travel the distance. To find the distance, d needs to be the isolated variable. r = d/t ⇒ d = rt Therefore, this formula can be substituted for the distance traveled in the written inequality. rt + 2 ≥ 10 Since the speed is asked in miles per hour and the tractor drove for half an hour, the given time is written as 12=0.5. 0.5r + 2 ≥ 10 Likewise in the second test, it is given that the tractor starts from 1.5 miles away and ends less than 20 miles away from the barn after 45 minutes. This means that the sum of the distance traveled and 1.5 miles has to be less than 20 miles. distance traveled + 1.5 < 20 Then, the formula for the distance traveled can be substituted in this inequality as well. rt + 1.5 < 20 In this case 45 minutes can be converted to hours using a conversion factor. 45 minutes are equal to 4560 = 0.75 hours. 0.75r + 1.5 < 20 Since the possible values for the speed are the values of r that satisfy both inequalities, these inequalities can be combined as a compound inequality with and. 0.5r + 2 ≥ 10 and 0.75r + 1.5 < 20 To solve this compound inequality, the properties of inequalities will be used. To solve the inequality on the left, first 2 will be subtracted from both sides of the inequality. Similarly, to solve the inequality on the right, 1.5 will be subtracted from both sides of the inequality. The compound inequality can be written with the found solutions. r ≥ 16 and r < 24.7 It should be noted that some compound inequalities that are written with and can be rewritten as follows. 16 ≤ r < 24.7 This means that the possible speeds of the tractor are greater than or equal to 16 miles per hour and less than 24.7 miles per hour.

On this case, since the inequality is strict, the solution set of the inequality r < 24.7 is made of the numbers to the left of 24.7. Since 24.7 is not included in the solution set, an open circle is used instead.

Since the compound inequality is written with and, its solution set is made of the numbers that satisfy both inequalities.

15 miles west<1/2d Since the distance is to the west of the volcano, it is written as a negative number. Also, since this value is negative, further distances from the volcano are less than -15. Therefore, the inequality has to be rewritten as follows. 1/2d < - 15 Similarly, it is given that the eastern flag is less than two thirds of the safe distance at 30 miles from the volcano. This can also be written as an inequality. Since the distances to the east are written with positive numbers, the inequality does not need to be modified. 30<2/3d These inequalities can be solved by using the Properties of Inequalities, a similar fashion to solving an equation. The first inequality is solved by multiplying by 2. The safe distance to the west is less than - 30 miles, or more than 30 miles. The second inequality can be solved by multiplying both sides by the reciprocal of 23, 32. The safe distance to the east is more than 45 miles. Since the safe distance can be either more than 30 miles west or more than 45 miles east, the compound inequality is written with or. d < -30 or  d > 45

Similarly, the graph of the solution set of d > 45 is made of every number to the right of 45, not including 45.

Finally, since it is written with or, the graph of the compound inequality is the combination of both solution sets and does not need to be adjusted or limited.

100 - 5d >0 Since Tearrik will eat cookies throughout this time, the values of d are the possible numbers of days in which he eats cookies. On the other hand, Tearrik's brother will save 3 cookies every day until he has at least 90 cookies. Then, Tearrik's brother will start eating his cookies. 3d ≥ 90 For both to eat cookies, the value d must satisfy both inequalities at the same time. This kind of compound inequality is written with and. 100-5d>0 and 3d ≥ 90 To solve this compound inequality, both inequalities must be solved. First, to solve the inequality on the left, 100 will be subtracted from both sides of the inequality. This means that Terrik will eat his cookies over the first 20 days. Since this inequality is strict, Tearrik will not reach the 20th day before he finishes his cookies. Then, the second inequality is solved by dividing both sides of the inequality by 3. This means that Terrik's brother will save his cookies for 30 days. Since the inequality is non-strict, he will start eating his cookies on the 30th day. Now the compound inequality can be rewritten using the solutions. d < 20 and d ≥ 30 This means that Tearrik will be able to eat cookies for 20 days, while his brother will save his cookies for 30 days before eating any of them. Therefore, they will not eat any cookies on the same day.

Similarly, the solution set of the inequality d ≥ 30 is made of the numbers greater than or equal to 30. Since thi inequality is not strict, the circle is closed.

The solution set of the compound inequality is made of the numbers that satisfy both inequalities at the same time. Since there are no such numbers, the compound inequality has no solution.

This confirms that the brothers will not eat cookies together.

F < 3272 Since the temperature is needed in degrees Celsius, the conversion formula will be substituted for F in the inequality above. Let C be the temperature in degrees Celsius. 9/5C + 32 < 3272 On the other hand, it is given that a pyrometer can measure temperatures as low as 973K. This can be written as an inequality, using K as the temperature in kelvins. It should be noted that kelvins are not degrees since it is an absolute scale. K ≥ 973 Substituting the conversion formula from the given table, the inequality can be rewritten in terms of degrees Celsius C. C + 273 ≥ 973 Since the question asks to describe the temperature range that both thermometers can measure, if either one can be used, the compound inequality is written with or. 9/5C + 32 < 3272 or C + 273 ≥ 973 Now, to solve this compound inequality, each inequality will be solved individually using the Properties of Inequalities. To solve the inequality on the left, first subtract 32 from both sides of the inequality. A similar process can be done to the inequality on the right. Therefore, the compound inequality can be rewritten with these solutions. C < 1800 or C ≥ 700

Similarly, the solution set of the inequality C ≥ 700 is made of every point to the right of 700, including 700.

Since the inequality is written with or, the solution set of the compound inequality is made of the numbers that satisfy either inequality. By combining the graphs it can be noted that every number is a solution to the compound inequality.

This means that any temperature can be measured using either of the thermometers.

2m Also, Diego's uncle will give Diego $500 dollars after finishing saving. This adds 500 to the amount after it is multiplied by 2. With this information, it is possible to write Diego's total amount of money as an expression. 2m + 500 Diego needs from 15 thousand dollars to 18 thousand dollars to afford a car he likes. Therefore, the total amount of money that Diego needs must lie between these values in order for him to be able to afford the car. This can be expressed as a compound inequality. 2m + 500 ≥ 15 000  and 2m + 500 ≤ 18 000 Since the inequalities are written with and, they can be rewritten as follows. 15 000 ≤ 2m + 500 ≤ 18 000

Since the expression is the same, the second inequality is solved following the same steps. The resulting inequality is obtained by combining these results, similar to how it was written in Part A. 7250 ≤ m ≤ 8750 Since the inequality is non-strict, Diego can start looking to buy the car after he saves some amount of money from $7250 to $8750, including these values. After he saves enough, he can stop saving money.

Similarly, the graph of the solution set of inequality m≤ 8750 is made of all the points to the left of 8750, including 8750.

Finally, the solution set of the compound inequality is made of every number that both graphs share. In this case, the overlapping space from 7250 to 8750, inclusive.

r = d/t In this formula, r is the rate or speed, d is the distance traveled, and t is the time duration of the movement. Since two time estimations are given, the time has to be the isolated quantity in the formula. It is given that it will take Vincenzo between 40 and 55 minutes to get home. Therefore, the time t in the formula has to be at least 40. This is the same as writing that t is greater than or equal to 40 minutes. t ≥ 40 minutes However, since the speed is given in miles per hour, the 40 minute interval must be written in terms of hours. This can be done using a conversion factor. 40 minutes* 1 hour/60 minutes = 2/3 hours The inequality can be rewritten using this information. t ≥ 2/3 hours Now the formula for the time in terms of the distance and the speed can be substituted into the inequality. It is given that Vincenzo will drive at 50 miles per hour. This means that r is equal to 50. d/50 ≥ 2/3 hours The same can be done with the upper limit of 55 minutes. The first to note is that the statement can be written as t is less than or equal to 55 minutes t ≤ 55 minutes Using the same conversion factor, the time can be rewritten in terms of hours. 55 minutes* 1 hour/60 minutes = 11/12 hours And using this time span, the inequality can be rewritten. t ≤ 11/12 hours Finally, the expression in terms of the distance can be substituted into the inequality. d/50 ≤ 11/12 hours Since the possible values for d satisfy both inequalities, the compound inequality is written with and. d/50 ≥ 2/3 and d/50 ≤ 11/12 Recall that compound inequalities written with and can be rewritten as follows. 2/3 ≤ d/50 ≤ 11/12 Finally, to find the possible values of d, each section of the compound inequality is multiplied by 50. This means that Vincenzo's house is somewhere between 33.3 and 45.8 miles away from the beach. To graph the solution set of this compound inequality, each point between 33.3 and 45.8 will be included. Note that the compound inequality is non-strict, so the endpoints are denoted with closed circles.

33.3 ≤ d ≤ 45.8 This compound inequality can be written with the word and. d ≥ 33.3 and d ≤ 45.8 The values of y are the values that fall outside of this compound inequality, so each individual inequality should be reversed. Note that because 33.3 and 45.8 are included in the original interval, these values cannot be included in the solution set for y. Since d is greater than or equal to 33.3 miles, then y must be less than 33.3 miles. y < 33.3 Similarly, since the values of d are less than or equal to 45.8 miles, y has to be greater than 45.8 miles. y > 45.8 The possible values of y are those that are either less than 33.3 or greater than 45.8. Therefore, the compound inequality is written with or. y < 33.3 or y > 45.8 The solution set of this compound inequality is made of every number left of 33.3 and right of 45.8.