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?
show it step by step
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?
show it step by step
Answers
GPT-4o mini
Sure! Let’s tackle each question step-by-step:
### Q1: Solve the equation
**Equation:**
\[ 3 = -2|14s - 5| + 3 \]
**Step 1: Subtract 3 from both sides**
\[ 0 = -2|14s - 5| \]
**Step 2: Divide by -2**
\[ 0 = |14s - 5| \]
This implies:
\[ 14s - 5 = 0 \]
\[ 14s = 5 \]
\[ s = \frac{5}{14} \]
**Answer:** \( s = \frac{5}{14} \)
### Q2 Part A: Identify the error in steps
**The original equation:**
\[ -14(20x - 8) = 15(10x) + 16 \]
**Step 1:** Correctly expands to:
\[ -280x + 112 = 150x + 16 \]
**Correct identification of error:**
The error appears at **Step 1** where it states: \[-5x + 2 = 2x + 16\]. The left side doesn't match.
### Q2 Part B: Solve the equation correctly
**Starting from the original equation:**
\[ -14(20x - 8) = 15(10x) + 16 \]
**Step 1: Expand both sides**
\[ -280x + 112 = 150x + 16 \]
**Step 2: Add \( 280x \) and subtract \( 16 \) from both sides**
\[ 112 - 16 = 150x + 280x \]
\[ 96 = 430x \]
**Step 3: Divide by 430**
\[ x = \frac{96}{430} = \frac{48}{215} \]
**Answer:** \( x = \frac{48}{215} \)
### Q3: Solve for \( h \)
**Formula:**
\[ S = 2r^2 + 2rh \]
**Step 1: Isolate terms with \( h \)**
\[ S - 2r^2 = 2rh \]
**Step 2: Solve for \( h \)**
\[ h = \frac{S - 2r^2}{2r} \]
**Answer:** \( h = \frac{S - 2r^2}{2r} \)
### Q4: Find the normal hourly rate
You worked 46 hours, with 6 hours over time. Let normal hourly rate = \( x \).
**Step 1:** Calculate the earnings:
\[ 40x + 6(1.2x) = 462.56 \]
\[ 40x + 7.2x = 462.56 \]
\[ 47.2x = 462.56 \]
**Step 2:** Divide:
\[ x = \frac{462.56}{47.2} \approx 9.79 \]
**Answer:** Normal hourly rate is approximately $9.79.
### Q5: Find the number \( n \)
**Equation:**
\[ 2n + 13 = 75 \]
**Step 1: Subtract 13 from both sides**
\[ 2n = 62 \]
**Step 2: Divide by 2**
\[ n = 31 \]
**Answer:** \( n = 31 \)
### Q6: Solve the inequality
**Inequality:**
\[ -7.5 < 2 - h \quad \text{or} \quad 6h + 5 > 71 \]
**For:** \( -7.5 < 2 - h \)
- Step 1: Rearranging gives \( h < 9.5 \)
**For:** \( 6h + 5 > 71 \)
- Step 1: Subtract 5 gives \( 6h > 66 \)
- Step 2: Divide by 6 gives \( h > 11 \)
**Final solution:**
\[ h < 9.5 \text{ or } h > 11 \]
**Graph:**
Numbers less than 9.5 and greater than 11 on a number line.
### Q7: Write an inequality for m
Assuming the graph shows a minimum wage, you will typically write:
\[ m \geq \text{minimum hourly wage value} \]
### Q8: Solve the equation
**Equation:**
\[ 280 = -8(-14 + x) \]
**Step 1: Distribute**
\[ 280 = 112 - 8x \]
**Step 2: Rearranging:**
\[ 280 - 112 = -8x \]
\[ 168 = -8x \]
**Step 3: Divide by -8:**
\[ x = -21 \]
**Answer:** \( x = -21 \)
### Q9: Solve \( 20x + 5y = 15 \) for \( y \)
**Step 1: Isolate \( 5y \)**
\[ 5y = 15 - 20x \]
**Step 2: Divide by 5**
\[ y = 3 - 4x \]
**Answer:** \( y = 3 - 4x \)
### Q10: Solve the formula
**Equation:**
\[ S = L - rL \]
**Step 1: Factor**
\[ S = L(1 - r) \]
**Step 2: Solve for \( r \)**
\[ 1 - r = \frac{S}{L} \]
\[ r = 1 - \frac{S}{L} \]
**List price of the shirt is $21.50 and \( S = L - rL\):**
Calculate discount percent with \( S \) being the final sale price. Assuming \( S \) is unknown here, replace \( S \) with \( L - rL \) to find:
**Example: \( r = 1 - \frac{\text{Sale Price}}{21.50} \)**
### Q11: Meeting time
**Equation:**
\[ 50h = 190 - 45h \]
**Step 1: Rearranging**
\[ 50h + 45h = 190 \]
\[ 95h = 190 \]
**Step 2: Divide**
\[ h = 2 \]
**Answer:** You will meet after 2 hours.
### Q12: Solve the inequality
**Inequality:**
\[ 4 - 2m > 7 - 3m \]
**Step 1: Rearranging gives**
\[ -2m + 3m > 7 - 4 \]
\[ m > 3 \]
**Answer:** \( m > 3 \)
### Q13: Graph the inequality
**Inequality:**
\[ x > 2 \]
**Graph:**
On a number line, an open circle at 2, shading to the right.
### Q14: Altitude of the plane
**Equation:**
\[ 21000 = 3400t + 600 \]
**Step 1: Rearranging gives**
\[ 21000 - 600 = 3400t \]
\[ 20400 = 3400t \]
**Step 2: Divide**
\[ t = \frac{20400}{3400} = 6 \]
**Answer:** It will take 6 minutes to reach 21,000 feet.
Let me know if you need further assistance!
### Q1: Solve the equation
**Equation:**
\[ 3 = -2|14s - 5| + 3 \]
**Step 1: Subtract 3 from both sides**
\[ 0 = -2|14s - 5| \]
**Step 2: Divide by -2**
\[ 0 = |14s - 5| \]
This implies:
\[ 14s - 5 = 0 \]
\[ 14s = 5 \]
\[ s = \frac{5}{14} \]
**Answer:** \( s = \frac{5}{14} \)
### Q2 Part A: Identify the error in steps
**The original equation:**
\[ -14(20x - 8) = 15(10x) + 16 \]
**Step 1:** Correctly expands to:
\[ -280x + 112 = 150x + 16 \]
**Correct identification of error:**
The error appears at **Step 1** where it states: \[-5x + 2 = 2x + 16\]. The left side doesn't match.
### Q2 Part B: Solve the equation correctly
**Starting from the original equation:**
\[ -14(20x - 8) = 15(10x) + 16 \]
**Step 1: Expand both sides**
\[ -280x + 112 = 150x + 16 \]
**Step 2: Add \( 280x \) and subtract \( 16 \) from both sides**
\[ 112 - 16 = 150x + 280x \]
\[ 96 = 430x \]
**Step 3: Divide by 430**
\[ x = \frac{96}{430} = \frac{48}{215} \]
**Answer:** \( x = \frac{48}{215} \)
### Q3: Solve for \( h \)
**Formula:**
\[ S = 2r^2 + 2rh \]
**Step 1: Isolate terms with \( h \)**
\[ S - 2r^2 = 2rh \]
**Step 2: Solve for \( h \)**
\[ h = \frac{S - 2r^2}{2r} \]
**Answer:** \( h = \frac{S - 2r^2}{2r} \)
### Q4: Find the normal hourly rate
You worked 46 hours, with 6 hours over time. Let normal hourly rate = \( x \).
**Step 1:** Calculate the earnings:
\[ 40x + 6(1.2x) = 462.56 \]
\[ 40x + 7.2x = 462.56 \]
\[ 47.2x = 462.56 \]
**Step 2:** Divide:
\[ x = \frac{462.56}{47.2} \approx 9.79 \]
**Answer:** Normal hourly rate is approximately $9.79.
### Q5: Find the number \( n \)
**Equation:**
\[ 2n + 13 = 75 \]
**Step 1: Subtract 13 from both sides**
\[ 2n = 62 \]
**Step 2: Divide by 2**
\[ n = 31 \]
**Answer:** \( n = 31 \)
### Q6: Solve the inequality
**Inequality:**
\[ -7.5 < 2 - h \quad \text{or} \quad 6h + 5 > 71 \]
**For:** \( -7.5 < 2 - h \)
- Step 1: Rearranging gives \( h < 9.5 \)
**For:** \( 6h + 5 > 71 \)
- Step 1: Subtract 5 gives \( 6h > 66 \)
- Step 2: Divide by 6 gives \( h > 11 \)
**Final solution:**
\[ h < 9.5 \text{ or } h > 11 \]
**Graph:**
Numbers less than 9.5 and greater than 11 on a number line.
### Q7: Write an inequality for m
Assuming the graph shows a minimum wage, you will typically write:
\[ m \geq \text{minimum hourly wage value} \]
### Q8: Solve the equation
**Equation:**
\[ 280 = -8(-14 + x) \]
**Step 1: Distribute**
\[ 280 = 112 - 8x \]
**Step 2: Rearranging:**
\[ 280 - 112 = -8x \]
\[ 168 = -8x \]
**Step 3: Divide by -8:**
\[ x = -21 \]
**Answer:** \( x = -21 \)
### Q9: Solve \( 20x + 5y = 15 \) for \( y \)
**Step 1: Isolate \( 5y \)**
\[ 5y = 15 - 20x \]
**Step 2: Divide by 5**
\[ y = 3 - 4x \]
**Answer:** \( y = 3 - 4x \)
### Q10: Solve the formula
**Equation:**
\[ S = L - rL \]
**Step 1: Factor**
\[ S = L(1 - r) \]
**Step 2: Solve for \( r \)**
\[ 1 - r = \frac{S}{L} \]
\[ r = 1 - \frac{S}{L} \]
**List price of the shirt is $21.50 and \( S = L - rL\):**
Calculate discount percent with \( S \) being the final sale price. Assuming \( S \) is unknown here, replace \( S \) with \( L - rL \) to find:
**Example: \( r = 1 - \frac{\text{Sale Price}}{21.50} \)**
### Q11: Meeting time
**Equation:**
\[ 50h = 190 - 45h \]
**Step 1: Rearranging**
\[ 50h + 45h = 190 \]
\[ 95h = 190 \]
**Step 2: Divide**
\[ h = 2 \]
**Answer:** You will meet after 2 hours.
### Q12: Solve the inequality
**Inequality:**
\[ 4 - 2m > 7 - 3m \]
**Step 1: Rearranging gives**
\[ -2m + 3m > 7 - 4 \]
\[ m > 3 \]
**Answer:** \( m > 3 \)
### Q13: Graph the inequality
**Inequality:**
\[ x > 2 \]
**Graph:**
On a number line, an open circle at 2, shading to the right.
### Q14: Altitude of the plane
**Equation:**
\[ 21000 = 3400t + 600 \]
**Step 1: Rearranging gives**
\[ 21000 - 600 = 3400t \]
\[ 20400 = 3400t \]
**Step 2: Divide**
\[ t = \frac{20400}{3400} = 6 \]
**Answer:** It will take 6 minutes to reach 21,000 feet.
Let me know if you need further assistance!