Could someone please show me how to do these problems? I would really appreciate any help you could give... There are examples in my lesson but I don't understand them, here are the examples~~
EXAMPLE~
Solve 3(x – 2) + 4 > 2(2x – 3).
First I'll multiply through and simplify; then I'll solve:
3(x – 2) + 4 > 2(2x – 3)
3x – 6 + 4 > 4x – 6
3x – 2 > 4x – 6
–2 > x – 6 (*)
4 > x
x < 4
EXAMPLE~
The velocity of an object fired directly upward is given by V = 80 – 32t, where t is in seconds. When will the velocity be between 32 and 64 feet per second?
32 < 80 – 32t < 64
32 – 80 < 80 – 80 – 32t < 64 – 80
–48 < –32t < –16
–48 / –32 > –32t / –32 > –16 / –32
1.5 > t > 0.5
0.5 < t < 1.5
The velocity will be between 32 and 64 feet per second between 0.5 seconds after launch and 1.5 seconds after launch.
1. Carlos spends at least 45 minutes each day on the Internet. Which inequality could be used to describe the amount of time, x, that Carlos spends on the Internet?
2. Amrita spends at most 2 hours watching television each night. Which inequality could be used to describe the amount of time, x, that Amrita spends watching television each night?
3. Joe expects to spend at least $8500 on a used car, but he can’t afford to spend more than $12,000. Which inequality could be used to describe the amount of money, x, that Joe may spend on a car?
4. Teenagers should sleep between 8 and 10 hours per night. Sarah typically sleeps 2 hours less than this per night. Which inequality describes the amount of time that Sarah sleeps?
5. Men need to intake between 2200 and 2800 calories daily. Women need 600 fewer calories than this. Write and solve an inequality to discover how many calories women should be taking in per day.
6. The velocity of an object fired directly upward is given by V = 20 – 9.8t, where t is time in seconds. When will the velocity be between 10 and 20 meters per second?
7. If you invest $2000 at a rate of 8% per year, how many years will it take to earn at least $1000 in interest?
8. If you invest $5000 for 10 years, which inequality describes the interest rate needed to earn at least $1500 of interest?
9. The velocity of an object fired directly upward is given by V = 35 – 9.8t, where t is time in seconds. Which inequality describes the velocity when t is between 0.5 and 1.5 seconds? (HINT: Remember that when you multiply an inequality by a negative number, you must switch the direction of the inequality.)
10. What is the least amount of principal you can invest at an 8% interest rate to earn at least $1000 in interest in 3 years? (Round your answer to the nearest dollar.)
3(x-2) + 4 > 2(2x-3).
3x-6 + 4 > 4x-6.
3x-4x > -6 + 2.
-x > -4.
X < 4.
1. X => 45 min.
2. X =< 2 hrs.
3. $8500 =< X < $12,000.
Sure, I'll be happy to help you with these problems step-by-step.
1. The amount of time Carlos spends on the Internet can be represented by the inequality x ≥ 45, since he spends at least 45 minutes each day.
2. The amount of time Amrita spends watching television can be represented by the inequality x ≤ 2, since she spends at most 2 hours each night.
3. Joe's possible spending on a used car can be represented by the inequality 8500 ≤ x ≤ 12000, since he expects to spend at least $8500 but not more than $12,000.
4. The amount of time Sarah sleeps can be represented by the inequality x ≤ 8 - 2, since she sleeps 2 hours less than the recommended 8 to 10 hours. Simplifying, x ≤ 6.
5. Women's calorie intake can be represented by the inequality x ≤ 2800 - 600, since they need 600 fewer calories than men. Simplifying, x ≤ 2200.
6. To find when the velocity is between 10 and 20 meters per second, we set up the inequality 10 < 20 - 9.8t < 20. Simplifying, we get 10 < -9.8t < 10. To solve for t, we divide by -9.8 (remembering to flip the inequality signs since we're dividing by a negative number): 10/(-9.8) > t > 10/(-9.8). Simplifying further, 1.02 > t > -1.02.
7. To find how many years it will take to earn at least $1000 in interest on a $2000 investment at a rate of 8% per year, we set up the inequality 1000 ≤ 2000 * 0.08 * t. Simplifying, 1000 ≤ 160t. Dividing by 160, we get t ≥ 6.25. So it will take at least 6.25 years to earn at least $1000 in interest.
8. To find the interest rate needed to earn at least $1500 of interest on a $5000 investment over 10 years, we set up the inequality 1500 ≤ 5000 * r * 10. Simplifying, 1500 ≤ 50000r. Dividing by 50000, we get r ≥ 0.03. So an interest rate of at least 3% is needed.
9. To describe the velocity when t is between 0.5 and 1.5 seconds, we set up the inequality 10 < 35 - 9.8t < 20. Simplifying, 10 < -9.8t + 35 < 20. Subtracting 35 from all sides, we get -25 < -9.8t < -15. Dividing by -9.8 (remembering to switch the inequality signs), we get 25/9.8 > t > 15/9.8. Simplifying further, 2.55 > t > 1.53.
10. To find the least amount of principal needed to earn at least $1000 in interest in 3 years at an 8% interest rate, we set up the inequality 1000 ≤ P * 0.08 * 3. Simplifying, 1000 ≤ 0.24P. Dividing by 0.24, we get P ≥ 4166.67. So the least amount of principal that should be invested is $4167 (rounded to the nearest dollar).
I hope this helps! Let me know if you have any other questions.
1. To solve this problem, we need to come up with an inequality that expresses the amount of time, x, that Carlos spends on the Internet. Since Carlos spends at least 45 minutes each day on the Internet, we can write the inequality as x ≥ 45. The greater than or equal to sign (≥) is used because Carlos spends at least 45 minutes, meaning he can spend more than 45 minutes if he wants.
2. To solve this problem, we need to come up with an inequality that expresses the amount of time, x, that Amrita spends watching television each night. Since Amrita spends at most 2 hours watching television, we can write the inequality as x ≤ 2. The less than or equal to sign (≤) is used because Amrita spends at most 2 hours, meaning she can spend less than 2 hours if she wants.
3. To solve this problem, we need to come up with an inequality that expresses the amount of money, x, that Joe may spend on a car. Since Joe expects to spend at least $8500 but can't afford to spend more than $12,000, we can write the inequality as 8500 ≤ x ≤ 12000. The range of values is represented by the less than or equal to sign (≤) and the greater than or equal to sign (≥).
4. To solve this problem, we need to come up with an inequality that describes the amount of time that Sarah sleeps. Since teenagers should sleep between 8 and 10 hours per night, and Sarah typically sleeps 2 hours less than this, we can write the inequality as 8 - 2 ≤ S ≤ 10 - 2, where S represents the amount of time that Sarah sleeps. The inequality represents the range of values from 6 to 8, inclusive.
5. To solve this problem, we need to come up with an inequality that describes the number of calories women should intake per day. Since men need to intake between 2200 and 2800 calories daily, and women need 600 fewer calories than this, we can write the inequality as 2200 - 600 ≤ C ≤ 2800 - 600, where C represents the number of calories women should intake per day. Simplifying the inequality gives us 1600 ≤ C ≤ 2200. The range of values is represented by the less than or equal to sign (≤) and the greater than or equal to sign (≥).
6. To solve this problem, we need to find the time when the velocity is between 10 and 20 meters per second. The given equation for velocity is V = 20 - 9.8t. Plugging in 10 for V, we get 10 = 20 - 9.8t. Solving for t gives us t = 1.02 seconds. Similarly, plugging in 20 for V, we get 20 = 20 - 9.8t, which gives us t = 0. This means the velocity is between 10 and 20 meters per second between t = 0 and t = 1.02 seconds.
7. To solve this problem, we need to find the number of years it takes to earn at least $1000 in interest when investing $2000 at an 8% interest rate. The formula for calculating interest is I = P * r * t, where I represents the interest, P represents the principal (initial investment), r represents the interest rate, and t represents the time in years. Plugging in the given values, we get 1000 = 2000 * 0.08 * t. Solving for t gives us t = 1000 / (2000 * 0.08), which simplifies to t = 6.25 years.
8. To solve this problem, we need to find the interest rate needed to earn at least $1500 of interest when investing $5000 for 10 years. Using the same formula as in the previous problem, we can write the inequality as 1500 ≤ 5000 * r * 10, where r represents the interest rate. Simplifying the inequality gives us 1500 ≤ 50000 * r. Dividing both sides by 50000 gives us 0.03 ≤ r. Therefore, the interest rate needed to earn at least $1500 of interest is 3% or greater.
9. To solve this problem, we need to find the inequality that describes the velocity when t is between 0.5 and 1.5 seconds. The given equation for velocity is V = 35 - 9.8t. Plugging in 0.5 for t, we get V = 35 - 9.8 * 0.5, which simplifies to V = 35 - 4.9, giving us V = 30.1. Plugging in 1.5 for t, we get V = 35 - 9.8 * 1.5, which simplifies to V = 35 - 14.7, giving us V = 20.3. Therefore, the inequality describing the velocity when t is between 0.5 and 1.5 seconds is 20.3 ≤ V ≤ 30.1.
10. To solve this problem, we need to find the least amount of principal that can be invested at an 8% interest rate to earn at least $1000 in interest in 3 years. Using the same formula as in the previous problems, we can write the equation as 1000 = P * 0.08 * 3, where P represents the principal. Simplifying the equation gives us 1000 = 0.24P. Dividing both sides by 0.24 gives us P = 4166.67. Rounding this to the nearest dollar gives us an answer of $4167. Therefore, the least amount of principal that can be invested is $4167.