Question
Q1 Solve the equation.
Solve 3=−2|14s−5|+3
Q2 Part A: A student solved the equation −14(20x−8)=15(10x)+16
as shown. In which step does an error appear first?
−14(20x−8)=15(10x)+16
Step 1: −5x+2=2x+16
Step 2: 2=−3x+16
Step 3: −14=−3x
Step 4: 143=x
Part B: Solve the equation correctly and show all of your work.
Q3 Solve the formula for h
.
Surface area of a cylinder: S=2r2+2rh
Q4 You are paid 1.2 times your normal hourly rate for each hour you work over 40 hours in a week. You work 46 hours this week and earn $462.56. What is your normal hourly rate?
Q5 Write and solve an equation to find the number n
. The sum of twice a number and 13 is 75.
Q6 Solve the inequality and graph the solution.
−7.5<2−h or 6h+5>71
Q7 The graph shows the hourly wage requirement m
(in dollars) for employees in a state. Write an inequality that represents the state’s hourly wage requirement.
Q8 What is the solution to the equation 280=−8(−14+x)
?
Q9 Solve 20x+5y=15
for y
.
Q10 The sale price S (in dollars) of an item is given by the formula S=L−rL
, where L
is the list price (in dollars) and r
is the percent of discount (in decimal form).
Solve the formula for r
.
The list price of the shirt is $21.50. What is the percent of discount?
Q11 You and your friend drive toward each other. The equation 50h=190−45h
represents the number h
of hours until you and your friend meet. After how many hours will you meet?
Q12 Solve the inequality
4−2m>7−3m
Q13 Graph the inequality
x>2
Q14 The altitude a
(in feet) of a plane t
minutes after takeoff is given by a=3400t+600
. How many minutes after takeoff is the plane at an altitude of 21,000 feet?
Pls show it step by step
Answers
GPT-4o mini
Sure! Let's tackle the questions step by step.
### Q1
Solve the equation:
\[
3 = -2|14s - 5| + 3
\]
Subtract 3 from both sides:
\[
0 = -2|14s - 5|
\]
Divide by -2:
\[
|14s - 5| = 0
\]
This implies:
\[
14s - 5 = 0
\]
So,
\[
14s = 5 \implies s = \frac{5}{14}
\]
### Q2
**Part A:** Identify the first error in the solution:
Starting with:
\[
-14(20x - 8) = 15(10x) + 16
\]
**Correct Steps:**
Multiply both sides effectively. Calculate:
\[
-280x + 112 = 150x + 16
\]
The student incorrectly arrived at:
\[
-5x + 2 = 2x + 16
\]
This is where the first error occurs since the correct equation should be \(-280x + 112 = 150x + 16\).
**Part B:** Solve the equation correctly:
Starting with:
\[
-280x + 112 = 150x + 16
\]
Add \(280x\) and subtract \(16\) from both sides:
\[
112 - 16 = 150x + 280x
\]
\[
96 = 430x
\]
Dividing both sides by 430:
\[
x = \frac{96}{430} = \frac{48}{215}
\]
### Q3
Solve the formula for \(h\):
The surface area of a cylinder is given by:
\[
S = 2r^2 + 2rh
\]
Isolate \(h\):
\[
S - 2r^2 = 2rh
\]
Divide both sides by \(2r\):
\[
h = \frac{S - 2r^2}{2r}
\]
### Q4
Let \(r\) be the normal hourly rate.
For 40 hours:
\[
40r
\]
For 6 hours overtime at 1.2 rate:
\[
6(1.2r) = 7.2r
\]
Thus,
\[
40r + 7.2r = 462.56
\]
Combine terms:
\[
47.2r = 462.56
\]
Divide by 47.2:
\[
r = \frac{462.56}{47.2} = 9.80
\]
### Q5
Let \(n\) be the number:
The equation is:
\[
2n + 13 = 75
\]
Subtract 13:
\[
2n = 62
\]
Divide by 2:
\[
n = 31
\]
### Q6
Solve the inequalities:
1. For \(-7.5 < 2 - h\):
\[
h < 2 + 7.5 \implies h < 9.5
\]
2. For \(6h + 5 > 71\):
\[
6h > 66 \implies h > 11
\]
Combined:
\(-7.5 < 2 - h\) or \(h > 11\)
### Q7
Typically, to express wage requirements:
Let’s assume \(m\) is the minimum wage.
A possible inequality could be:
\[
m \geq x
\]
Where \(x\) is defined based upon your graph specifics.
### Q8
Starting with:
\[
280 = -8(-14 + x)
\]
Distribute on the right:
\[
280 = 112 - 8x
\]
Add \(8x\) and \(280 - 112\):
\[
168 = -8x
\]
Divide by -8:
\[
x = -21
\]
### Q9
Start with:
\[
20x + 5y = 15
\]
To solve for \(y\):
\[
5y = 15 - 20x
\]
Divide by 5:
\[
y = 3 - 4x
\]
### Q10
Start with:
\[
S = L - rL
\]
Factor out \(L\):
\[
S = L(1 - r)
\]
Divide both sides by \(L\):
\[
\frac{S}{L} = 1 - r
\]
Thus,
\[
r = 1 - \frac{S}{L}
\]
List price \(L = 21.50\), sale price \(S\) is calcualted based on your next part (context needed).
### Q11
Given:
\[
50h = 190 - 45h
\]
Add \(45h\) both sides:
\[
95h = 190
\]
Divide by 95:
\[
h = 2
\]
### Q12
\[
4 - 2m > 7 - 3m
\]
Add \(3m\):
\[
4 + m > 7
\]
Subtract \(4\):
\[
m > 3
\]
### Q13
Graph the inequality \(x > 2\):
Draw a number line and mark an open circle at 2, shading to the right.
### Q14
Given:
\[
a = 3400t + 600 \]
Set to the altitude of 21,000:
\[
21000 = 3400t + 600
\]
Subtract 600:
\[
20400 = 3400t
\]
Divide by 3400:
\[
t = 6
\]
That's it! If you have further questions or need more details on any step, feel free to ask!
### Q1
Solve the equation:
\[
3 = -2|14s - 5| + 3
\]
Subtract 3 from both sides:
\[
0 = -2|14s - 5|
\]
Divide by -2:
\[
|14s - 5| = 0
\]
This implies:
\[
14s - 5 = 0
\]
So,
\[
14s = 5 \implies s = \frac{5}{14}
\]
### Q2
**Part A:** Identify the first error in the solution:
Starting with:
\[
-14(20x - 8) = 15(10x) + 16
\]
**Correct Steps:**
Multiply both sides effectively. Calculate:
\[
-280x + 112 = 150x + 16
\]
The student incorrectly arrived at:
\[
-5x + 2 = 2x + 16
\]
This is where the first error occurs since the correct equation should be \(-280x + 112 = 150x + 16\).
**Part B:** Solve the equation correctly:
Starting with:
\[
-280x + 112 = 150x + 16
\]
Add \(280x\) and subtract \(16\) from both sides:
\[
112 - 16 = 150x + 280x
\]
\[
96 = 430x
\]
Dividing both sides by 430:
\[
x = \frac{96}{430} = \frac{48}{215}
\]
### Q3
Solve the formula for \(h\):
The surface area of a cylinder is given by:
\[
S = 2r^2 + 2rh
\]
Isolate \(h\):
\[
S - 2r^2 = 2rh
\]
Divide both sides by \(2r\):
\[
h = \frac{S - 2r^2}{2r}
\]
### Q4
Let \(r\) be the normal hourly rate.
For 40 hours:
\[
40r
\]
For 6 hours overtime at 1.2 rate:
\[
6(1.2r) = 7.2r
\]
Thus,
\[
40r + 7.2r = 462.56
\]
Combine terms:
\[
47.2r = 462.56
\]
Divide by 47.2:
\[
r = \frac{462.56}{47.2} = 9.80
\]
### Q5
Let \(n\) be the number:
The equation is:
\[
2n + 13 = 75
\]
Subtract 13:
\[
2n = 62
\]
Divide by 2:
\[
n = 31
\]
### Q6
Solve the inequalities:
1. For \(-7.5 < 2 - h\):
\[
h < 2 + 7.5 \implies h < 9.5
\]
2. For \(6h + 5 > 71\):
\[
6h > 66 \implies h > 11
\]
Combined:
\(-7.5 < 2 - h\) or \(h > 11\)
### Q7
Typically, to express wage requirements:
Let’s assume \(m\) is the minimum wage.
A possible inequality could be:
\[
m \geq x
\]
Where \(x\) is defined based upon your graph specifics.
### Q8
Starting with:
\[
280 = -8(-14 + x)
\]
Distribute on the right:
\[
280 = 112 - 8x
\]
Add \(8x\) and \(280 - 112\):
\[
168 = -8x
\]
Divide by -8:
\[
x = -21
\]
### Q9
Start with:
\[
20x + 5y = 15
\]
To solve for \(y\):
\[
5y = 15 - 20x
\]
Divide by 5:
\[
y = 3 - 4x
\]
### Q10
Start with:
\[
S = L - rL
\]
Factor out \(L\):
\[
S = L(1 - r)
\]
Divide both sides by \(L\):
\[
\frac{S}{L} = 1 - r
\]
Thus,
\[
r = 1 - \frac{S}{L}
\]
List price \(L = 21.50\), sale price \(S\) is calcualted based on your next part (context needed).
### Q11
Given:
\[
50h = 190 - 45h
\]
Add \(45h\) both sides:
\[
95h = 190
\]
Divide by 95:
\[
h = 2
\]
### Q12
\[
4 - 2m > 7 - 3m
\]
Add \(3m\):
\[
4 + m > 7
\]
Subtract \(4\):
\[
m > 3
\]
### Q13
Graph the inequality \(x > 2\):
Draw a number line and mark an open circle at 2, shading to the right.
### Q14
Given:
\[
a = 3400t + 600 \]
Set to the altitude of 21,000:
\[
21000 = 3400t + 600
\]
Subtract 600:
\[
20400 = 3400t
\]
Divide by 3400:
\[
t = 6
\]
That's it! If you have further questions or need more details on any step, feel free to ask!