Could someone please show me how to do these problems? or just show me how to do 1 or 2? 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.
Questions~~
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.)
The two examples you have are pretty straight-forward.
You will just have to practise and practise until you can do those.
I will do #6
10 < (20-9.8t) < 20
subtract 20 from each part
-10 < -9.8t < 0
divide each part by -9.8, remember to reverse the inequality sign
10/9.8 > t > 0
or
0 < t < appr 1.02
you can always test them by picking numbers in that domain, and then choosing some numbers outside the domain to see what the velocity is
#9 is done the same way
#7and 8 are the same type.
they both use the compound interest formula
amount = present value(1+i)^n, where i the periodic intersest rate and n is the number of interest periods
I will do #7, you try #8
7. 2000(1 + .08)^n > 3000
divide by +2000, inequality sign stays the same
1.08^n > 1.5
take logs of both sides,
log(1.08^n) > log 1.5
n log 1.08 > log 1.5
divide both sides by log1.08, which is a positive number
n > log 1.5/log 1.08
n > 5.27
So it will take about 5 1/4 years to earn at least $1000 interest (or the amount is $3000)
Sure, I'd be happy to help you with these problems! Let's go through each one step by step:
6. The velocity of an object fired directly upward is given by V = 20 - 9.8t, where t is time in seconds. We want to find when the velocity is between 10 and 20 meters per second.
To do this, we set up the following inequality:
10 < 20 - 9.8t < 20
Now let's solve for t:
10 < 20 - 9.8t [subtract 20 from both sides]
-10 < -9.8t [divide by -9.8, and remember when you divide by a negative number, you must switch the direction of the inequality]
1.02 > t
So the velocity will be between 10 and 20 meters per second when t is less than 1.02 seconds.
7. If you invest $2000 at a rate of 8% per year, we want to find how many years it will take to earn at least $1000 in interest.
Let's set up an inequality to represent this situation:
1000 ≤ 2000 * 0.08 * t
Simplifying this inequality:
1000 ≤ 160t
1000/160 ≤ t
6.25 ≤ t
So it will take at least 6.25 years to earn $1000 in interest.
8. If you invest $5000 for 10 years, we want to find the inequality that describes the interest rate needed to earn at least $1500 of interest.
Let's set up the inequality:
1500 ≤ 5000 * r * 10
Simplifying this inequality:
1500 ≤ 50000r
1500/50000 ≤ r
0.03 ≤ r
So the interest rate needed to earn at least $1500 of interest is 0.03 or 3%.
9. The velocity of an object fired directly upward is given by V = 35 - 9.8t, where t is time in seconds. We want to find the inequality that describes the velocity when t is between 0.5 and 1.5 seconds.
Let's substitute the values of t into the equation and write the inequality:
35 - 9.8(0.5) ≤ V ≤ 35 - 9.8(1.5)
Simplifying the inequality:
30.1 ≤ V ≤ 20.5
So the velocity will be between 20.5 and 30.1 when t is between 0.5 and 1.5 seconds.
10. To find the least amount of principal you can invest at an 8% interest rate to earn at least $1000 in interest in 3 years, we can use the simple interest formula: Interest = Principal * Rate * Time.
Let's set up the inequality:
1000 ≤ P * 0.08 * 3
Simplifying the inequality:
1000 ≤ 0.24P
1000/0.24 ≤ P
Rounding to the nearest dollar:
4166.67 ≤ P
So the least amount of principal you can invest is $4167 to earn at least $1000 in interest in 3 years.
I hope this helps! Let me know if you have any further questions.
Sure, I can help you with a few of these problems. Let's start with problem 6.
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?
To solve this, we need to find the values of t that make the velocity between 10 and 20.
So, we can set up the inequality as follows:
10 ≤ 20 - 9.8t ≤ 20
Next, we will solve for t:
10 ≤ 20 - 9.8t ≤ 20
Subtract 20 from all parts of the inequality:
-10 ≤ -9.8t ≤ 0
Divide all parts of the inequality by -9.8 (remember to flip the inequality signs when dividing by a negative number):
-10 / -9.8 ≥ -9.8t / -9.8 ≥ 0 / -9.8
Simplifying:
1.02 ≥ t ≥ 0
Therefore, the velocity will be between 10 and 20 meters per second when t is between 0 seconds and 1.02 seconds.
Now, let's move on to problem 7.
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?
To solve this problem, we can use the formula for compound interest, which is:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, including interest
P = the principal amount (initial investment)
r = annual interest rate (expressed as a decimal)
n = number of times that interest is compounded per year (we'll assume it's compounded once per year, so n = 1)
t = number of years
We want to find the value of t that makes A at least $1000.
So, we can set up the equation as follows:
A = P(1 + r/n)^(nt)
1000 = 2000(1 + 0.08/1)^(1*t)
Divide both sides of the equation by 2000:
0.5 = (1.08)^t
Take the natural logarithm of both sides of the equation:
ln(0.5) = ln((1.08)^t)
Using the logarithmic property that ln(a^b) = b * ln(a):
ln(0.5) = t * ln(1.08)
Divide both sides of the equation by ln(1.08):
t = ln(0.5) / ln(1.08)
Using a calculator, you can find that t ≈ 8.36 years.
Therefore, it will take approximately 8.36 years to earn at least $1000 in interest if you invest $2000 at a rate of 8% per year.
I hope this helps! Let me know if you have any further questions.