could someone help me revise these Inequalities problems? my answers are under the question...

For questions, #3 #4 and #5 my teacher said "equals as well?"
For question #8 my teacher said "where's the inequality sign?"

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?

The cost of Joe’s car is x. So it is greater than $8500 but less than $12000.

$8500<x<12000

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?

The number of hours that Sarah is x. Which is 2 hours less than the typical hours of sleep, 8,

8-2=6

6<x<8



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.

Women’s requirements are x. They need 600 fewer calories than the minimum of a man’s requirements which are between 2200 and 2800.

1600<x<2200

8. If you invest $5000 for 10 years, which inequality describes the interest rate needed to earn at least $1500 of interest?

Interest rate is 3% per year, put 3% in decimal form ~ 3/100=0.03
5000*0.03*10=$1500
The answer is ~ interest rate needed to earn at least $1500 of interest is 3%

Thank you so much!!

at least means >= (not less)

at most means <= (not more)

#3: 8500 <= x <= 12000
#4 between usually includes the end points. So 6 <= x <= 8
#5: same idea

#8: Using simple interest,
5000(1+10r) >= 1500
1+10r >= 1.30
10r >= 0.3
r >= 0.03
so, at least 3%

3. To describe the amount of money, x, that Joe may spend on a car, we can use the inequality:

$8500 < x < $12000

4. To describe the amount of time that Sarah sleeps, we can use the inequality:

6 < x < 8

5. To determine the number of calories women should be taking in per day, we can use the inequality:

1600 < x < 2200

8. To earn at least $1500 of interest by investing $5000 for 10 years, we need to find the interest rate. Let's denote the interest rate as r. The equation to find the interest earned is:

Interest = Principal * Rate * Time

Therefore, we have the equation:

$1500 = $5000 * r * 10

Simplifying,

r = $1500 / ($5000 * 10)

r = $1500 / $50000

r = 0.03

So, the required interest rate needed is 0.03 or 3% per year.

To revise these inequality problems, let's make sure that our answers match the questions and include all the necessary information. I will go through each problem and provide a revised explanation.

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?

To revise the answer, we need to include the correct inequality symbols. Since Joe wants to spend at least $8500, we use the greater than or equal to symbol (≥). Similarly, since he can't afford to spend more than $12,000, we use the less than or equal to symbol (≤). So the correct inequality would be:
$8500 ≤ x ≤ $12000

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?

To revise the answer, we need to correctly represent the range of hours that Sarah sleeps. If Sarah sleeps 2 hours less than the typical range of 8 and 10 hours, we subtract 2 from both the lower and upper bounds. Therefore, the correct inequality would be:
6 ≤ x ≤ 8

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.

To revise the answer, we need to correctly represent the range of calories that women should be taking in. Since women need 600 fewer calories than the lower and upper bounds for men, we subtract 600 from both. Therefore, the correct inequality would be:
1600 ≤ x ≤ 2200

8. If you invest $5000 for 10 years, which inequality describes the interest rate needed to earn at least $1500 of interest?

To revise the answer, we need to express the inequality in terms of the interest rate. We know the formula for calculating interest is: Interest = Principal × Rate × Time. In this case, the principal is $5000, the interest is $1500, and the time is 10 years. Plugging in these values, we can solve for the rate:
$1500 = $5000 × Rate × 10
Simplifying, we get:
Rate = $1500 / ($5000 × 10)
Rate = 0.03 or 3/100, which corresponds to 3% in decimal form.

Therefore, the correct inequality would be:
Rate ≥ 0.03 or Rate ≥ 3%