Asked by I_EAT_EVERYDAY
Solve the inequality 6x + 1 < 5x − 8
x__|___
x__|___
Answers
Answered by
I_EAT_EVERYDAY
Shantel offers online tutorial services. She charges $25.00 an hour per student. Her budget tracker reflects a total of $3,200.00 monthly expenses. How many hours must Shantel work each month to earn a monthly profit of at least $600.00?
at ____ least
hours
at ____ least
hours
Answered by
I_EAT_EVERYDAY
La’Shondra solves the inequality 6+3x≤24
. She begins by subtracting 6 from both sides of the inequality and then divides both sides of the equivalent inequality by a number. What is the number?
. She begins by subtracting 6 from both sides of the inequality and then divides both sides of the equivalent inequality by a number. What is the number?
Answered by
I_EAT_EVERYDAY
Solve the inequality Rx > 10, where R is a number greater than 0.
x___|____ over R
x___|____ over R
Answered by
I_EAT_EVERYDAY
Solve the inequality Ay+3<12, where A is a number greater than 0
y___|___ over A
y___|___ over A
Answered by
I_EAT_EVERYDAY
Jorelle is planning a 5-day vacation. The trip will cost him $1,300.00 for airfare, food, and itineraries. He plans to stay in a hotel room for $80.00 per night. He only has $950.00 in savings, but he earns $25.00 per hour from his online tutoring services. Create a linear inequality in one variable that would best translate the problem. Determine the minimum number of hours Jorelle must tutor so he can have enough money for the trip.
A). 950+25x≤1,300; at most 14 hours
B). 950+25x≤1,700; at most 30 hours
C). 950+25x≥1,300; at least 14 hours
D). 950+25x≥1,700; at least 30 hours
A). 950+25x≤1,300; at most 14 hours
B). 950+25x≤1,700; at most 30 hours
C). 950+25x≥1,300; at least 14 hours
D). 950+25x≥1,700; at least 30 hours
Answered by
I_EAT_EVERYDAY
Mary Ann solves the inequality −3x≥12 . Justify the steps Mary Ann takes to find the solution set.
Options:
A). Mary Ann adds −3 to both sides of the inequality to get x≥9.
B). Mary Ann adds −3 to both sides of the inequality and changes the inequality symbol to ≤ to get x≤9.
C). Mary Ann divides both sides of the inequality by −3 and changes the inequality symbol to ≤ to get x≤−4.
D). Mary Ann divides both sides of the inequality by −3 to get x≥−4.
Options:
A). Mary Ann adds −3 to both sides of the inequality to get x≥9.
B). Mary Ann adds −3 to both sides of the inequality and changes the inequality symbol to ≤ to get x≤9.
C). Mary Ann divides both sides of the inequality by −3 and changes the inequality symbol to ≤ to get x≤−4.
D). Mary Ann divides both sides of the inequality by −3 to get x≥−4.
Answered by
I_EAT_EVERYDAY
Kirk solves the inequality 5 < 12y − 7 . Which statement explains the steps he uses to find the solution set of the inequality?
A). Kirk adds −7 to both sides of the inequality and then divides both sides of the equivalent inequality by 12 to get (−1 over 6) < y.
B). Kirk adds 7 to both sides of the inequality and then multiplies both sides of the equivalent inequality by 12 to get 144<y.
C). Kirk adds 7 to both sides of the inequality and then divides both sides of the equivalent inequality by 12 to get 1<y.
D). Kirk adds 7 to both sides of the inequality, then divides both sides of the equivalent inequality by 12 and reverses the inequality symbol to get 1>y.
A). Kirk adds −7 to both sides of the inequality and then divides both sides of the equivalent inequality by 12 to get (−1 over 6) < y.
B). Kirk adds 7 to both sides of the inequality and then multiplies both sides of the equivalent inequality by 12 to get 144<y.
C). Kirk adds 7 to both sides of the inequality and then divides both sides of the equivalent inequality by 12 to get 1<y.
D). Kirk adds 7 to both sides of the inequality, then divides both sides of the equivalent inequality by 12 and reverses the inequality symbol to get 1>y.
Answered by
I_EAT_EVERYDAY
Which of the following shows how to solve the linear inequality with a letter coefficient Hy < 7 , where H is a number greater than 0?
A). y < 7 over H
B). y < 7
C). y < 7H
Hy < 7
A). y < 7 over H
B). y < 7
C). y < 7H
Hy < 7
Answered by
I_EAT_EVERYDAY
Which term in the following inequality is a letter coefficient? Bx + 72 < 12
Answered by
GPT-4o mini
To solve the inequality \( 6x + 1 < 5x - 8 \), we can start by isolating \( x \).
1. Subtract \( 5x \) from both sides:
\[
6x - 5x + 1 < -8
\]
This simplifies to:
\[
x + 1 < -8
\]
2. Next, subtract \( 1 \) from both sides:
\[
x < -8 - 1
\]
This simplifies to:
\[
x < -9
\]
So the solution to the inequality is:
\[
x < -9
\]
In interval notation, this would be expressed as:
\[
(-\infty, -9)
\]
Thus, the final answer is:
\[
x < -9
\] or \( (-\infty, -9) \).
1. Subtract \( 5x \) from both sides:
\[
6x - 5x + 1 < -8
\]
This simplifies to:
\[
x + 1 < -8
\]
2. Next, subtract \( 1 \) from both sides:
\[
x < -8 - 1
\]
This simplifies to:
\[
x < -9
\]
So the solution to the inequality is:
\[
x < -9
\]
In interval notation, this would be expressed as:
\[
(-\infty, -9)
\]
Thus, the final answer is:
\[
x < -9
\] or \( (-\infty, -9) \).
Answered by
GPT-4o mini
To find out how many hours Shantel must work each month to earn a monthly profit of at least $600, we first need to establish the total income she needs to cover both her expenses and her desired profit.
1. **Total expenses**: $3,200.00
2. **Desired profit**: $600.00
The total amount she needs to earn each month is:
\[
\text{Total earnings required} = \text{Total expenses} + \text{Desired profit} = 3200 + 600 = 3800
\]
3. **Charge per hour**: $25.00
Let \( h \) be the number of hours she works in a month. The total income from her tutoring services can be expressed as:
\[
\text{Income} = \text{Charge per hour} \times h = 25h
\]
4. **Set up the inequality**:
To find the minimum hours needed to meet her income goal, we can set up the following inequality:
\[
25h \geq 3800
\]
5. **Solve for \( h \)**:
\[
h \geq \frac{3800}{25}
\]
Calculating the right side:
\[
h \geq 152
\]
Thus, Shantel must work at **least** 152 hours each month to earn a monthly profit of at least $600.
The answer is:
\[
\text{at least } 152 \text{ hours}
\]
1. **Total expenses**: $3,200.00
2. **Desired profit**: $600.00
The total amount she needs to earn each month is:
\[
\text{Total earnings required} = \text{Total expenses} + \text{Desired profit} = 3200 + 600 = 3800
\]
3. **Charge per hour**: $25.00
Let \( h \) be the number of hours she works in a month. The total income from her tutoring services can be expressed as:
\[
\text{Income} = \text{Charge per hour} \times h = 25h
\]
4. **Set up the inequality**:
To find the minimum hours needed to meet her income goal, we can set up the following inequality:
\[
25h \geq 3800
\]
5. **Solve for \( h \)**:
\[
h \geq \frac{3800}{25}
\]
Calculating the right side:
\[
h \geq 152
\]
Thus, Shantel must work at **least** 152 hours each month to earn a monthly profit of at least $600.
The answer is:
\[
\text{at least } 152 \text{ hours}
\]
Answered by
GPT-4o mini
Let's start solving the inequality given by La'Shondra:
\[
6 + 3x \leq 24
\]
1. **Subtract 6 from both sides**:
\[
3x \leq 24 - 6
\]
This simplifies to:
\[
3x \leq 18
\]
2. **Divide both sides by 3**:
\[
x \leq \frac{18}{3}
\]
This simplifies to:
\[
x \leq 6
\]
So, La'Shondra divides both sides of the inequality by **3**.
In summary, the number she divides by is:
\[
\text{3}
\]
\[
6 + 3x \leq 24
\]
1. **Subtract 6 from both sides**:
\[
3x \leq 24 - 6
\]
This simplifies to:
\[
3x \leq 18
\]
2. **Divide both sides by 3**:
\[
x \leq \frac{18}{3}
\]
This simplifies to:
\[
x \leq 6
\]
So, La'Shondra divides both sides of the inequality by **3**.
In summary, the number she divides by is:
\[
\text{3}
\]
Answered by
GPT-4o mini
To solve the inequality \( Rx > 10 \) where \( R \) is a number greater than 0, we can isolate \( x \) by dividing both sides of the inequality by \( R \).
Given \( R > 0 \), we proceed as follows:
1. Divide both sides by \( R \):
\[
\frac{Rx}{R} > \frac{10}{R}
\]
This simplifies to:
\[
x > \frac{10}{R}
\]
So the solution to the inequality is:
\[
x > \frac{10}{R}
\]
In interval notation, this notation indicates that \( x \) is greater than \( \frac{10}{R} \):
\[
\left( \frac{10}{R}, \infty \right)
\]
Thus, filling in the blank gives us:
\[
x > \frac{10}{R}
\]
Given \( R > 0 \), we proceed as follows:
1. Divide both sides by \( R \):
\[
\frac{Rx}{R} > \frac{10}{R}
\]
This simplifies to:
\[
x > \frac{10}{R}
\]
So the solution to the inequality is:
\[
x > \frac{10}{R}
\]
In interval notation, this notation indicates that \( x \) is greater than \( \frac{10}{R} \):
\[
\left( \frac{10}{R}, \infty \right)
\]
Thus, filling in the blank gives us:
\[
x > \frac{10}{R}
\]
Answered by
GPT-4o mini
To solve the inequality \( Ay + 3 < 12 \), where \( A \) is a number greater than 0, we will first isolate \( y \).
1. **Subtract 3 from both sides**:
\[
Ay < 12 - 3
\]
This simplifies to:
\[
Ay < 9
\]
2. **Divide both sides by \( A \)**:
Since \( A > 0 \), dividing by \( A \) does not change the direction of the inequality:
\[
\frac{Ay}{A} < \frac{9}{A}
\]
This simplifies to:
\[
y < \frac{9}{A}
\]
So, the solution to the inequality is:
\[
y < \frac{9}{A}
\]
In interval notation, this can be expressed as:
\[
(-\infty, \frac{9}{A})
\]
Thus, filling in the blank gives us:
\[
y < \frac{9}{A}
\]
1. **Subtract 3 from both sides**:
\[
Ay < 12 - 3
\]
This simplifies to:
\[
Ay < 9
\]
2. **Divide both sides by \( A \)**:
Since \( A > 0 \), dividing by \( A \) does not change the direction of the inequality:
\[
\frac{Ay}{A} < \frac{9}{A}
\]
This simplifies to:
\[
y < \frac{9}{A}
\]
So, the solution to the inequality is:
\[
y < \frac{9}{A}
\]
In interval notation, this can be expressed as:
\[
(-\infty, \frac{9}{A})
\]
Thus, filling in the blank gives us:
\[
y < \frac{9}{A}
\]
Answered by
GPT-4o mini
To solve the problem, let's first determine how much money Jorelle needs for his trip in total.
1. **Cost of the trip**: $1,300.00
2. **Cost of hotel for 5 nights**:
\[
\text{Hotel cost} = 5 \text{ nights} \times 80 \text{ dollars/night} = 400 \text{ dollars}
\]
3. **Total cost of the trip**:
\[
\text{Total trip cost} = \text{Airfare, food, etc.} + \text{Hotel cost} = 1,300 + 400 = 1,700 \text{ dollars}
\]
Jorelle has $950.00 in savings. Let \( x \) be the number of hours he needs to tutor. He earns $25.00 per hour from tutoring, so his total money from tutoring after working \( x \) hours will be \( 25x \).
Now, we want to create a linear inequality that represents the situation of having enough money for the trip:
**Inequality**:
\[
950 + 25x \geq 1,700
\]
This translates to Jorelle needing at least $1,700.00 to cover the total cost of the trip.
Now, let's solve the inequality for \( x \):
1. **Subtract 950** from both sides:
\[
25x \geq 1,700 - 950
\]
\[
25x \geq 750
\]
2. **Divide by 25**:
\[
x \geq \frac{750}{25}
\]
\[
x \geq 30
\]
Thus, Jorelle must tutor for **at least 30 hours** to have enough money for the trip.
Therefore, the correct answer is:
**D)** \( 950 + 25x \geq 1,700; \) at least **30 hours**.
1. **Cost of the trip**: $1,300.00
2. **Cost of hotel for 5 nights**:
\[
\text{Hotel cost} = 5 \text{ nights} \times 80 \text{ dollars/night} = 400 \text{ dollars}
\]
3. **Total cost of the trip**:
\[
\text{Total trip cost} = \text{Airfare, food, etc.} + \text{Hotel cost} = 1,300 + 400 = 1,700 \text{ dollars}
\]
Jorelle has $950.00 in savings. Let \( x \) be the number of hours he needs to tutor. He earns $25.00 per hour from tutoring, so his total money from tutoring after working \( x \) hours will be \( 25x \).
Now, we want to create a linear inequality that represents the situation of having enough money for the trip:
**Inequality**:
\[
950 + 25x \geq 1,700
\]
This translates to Jorelle needing at least $1,700.00 to cover the total cost of the trip.
Now, let's solve the inequality for \( x \):
1. **Subtract 950** from both sides:
\[
25x \geq 1,700 - 950
\]
\[
25x \geq 750
\]
2. **Divide by 25**:
\[
x \geq \frac{750}{25}
\]
\[
x \geq 30
\]
Thus, Jorelle must tutor for **at least 30 hours** to have enough money for the trip.
Therefore, the correct answer is:
**D)** \( 950 + 25x \geq 1,700; \) at least **30 hours**.
Answered by
GPT-4o mini
To solve the inequality \( -3x \geq 12 \), Mary Ann must perform a series of steps while keeping in mind the property of inequalities that states: **When you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality symbol.**
Starting with the inequality:
\[
-3x \geq 12
\]
1. **Divide both sides by -3**:
Since we are dividing both sides by a negative number, we must reverse the inequality symbol:
\[
\frac{-3x}{-3} \leq \frac{12}{-3}
\]
This simplifies to:
\[
x \leq -4
\]
### Conclusion:
The correct step Mary Ann takes is:
**C). Mary Ann divides both sides of the inequality by −3 and changes the inequality symbol to ≤ to get \( x \leq -4 \).**
Starting with the inequality:
\[
-3x \geq 12
\]
1. **Divide both sides by -3**:
Since we are dividing both sides by a negative number, we must reverse the inequality symbol:
\[
\frac{-3x}{-3} \leq \frac{12}{-3}
\]
This simplifies to:
\[
x \leq -4
\]
### Conclusion:
The correct step Mary Ann takes is:
**C). Mary Ann divides both sides of the inequality by −3 and changes the inequality symbol to ≤ to get \( x \leq -4 \).**
Answered by
GPT-4o mini
To solve the inequality \( 5 < 12y - 7 \), let's break down the steps that Kirk would correctly take:
1. **Start with the original inequality**:
\[
5 < 12y - 7
\]
2. **Add 7 to both sides** to isolate the term with \( y \):
\[
5 + 7 < 12y
\]
This simplifies to:
\[
12 < 12y
\]
3. **Now, divide both sides by 12**:
\[
\frac{12}{12} < y
\]
This simplifies to:
\[
1 < y
\]
or equivalently:
\[
y > 1
\]
### Conclusion:
The correct statement explaining the steps that Kirk uses is:
**C). Kirk adds 7 to both sides of the inequality and then divides both sides of the equivalent inequality by 12 to get \( 1 < y \).**
1. **Start with the original inequality**:
\[
5 < 12y - 7
\]
2. **Add 7 to both sides** to isolate the term with \( y \):
\[
5 + 7 < 12y
\]
This simplifies to:
\[
12 < 12y
\]
3. **Now, divide both sides by 12**:
\[
\frac{12}{12} < y
\]
This simplifies to:
\[
1 < y
\]
or equivalently:
\[
y > 1
\]
### Conclusion:
The correct statement explaining the steps that Kirk uses is:
**C). Kirk adds 7 to both sides of the inequality and then divides both sides of the equivalent inequality by 12 to get \( 1 < y \).**
Answered by
GPT-4o mini
To solve the linear inequality \( Hy < 7 \) where \( H \) is a number greater than 0, we need to isolate \( y \).
Given the inequality:
\[
Hy < 7
\]
We will divide both sides by \( H \). Since \( H > 0 \), the direction of the inequality does not change.
1. **Divide both sides by H**:
\[
\frac{Hy}{H} < \frac{7}{H}
\]
This simplifies to:
\[
y < \frac{7}{H}
\]
### Conclusion:
The correct way to represent the solution is:
**A). \( y < \frac{7}{H} \)**.
Given the inequality:
\[
Hy < 7
\]
We will divide both sides by \( H \). Since \( H > 0 \), the direction of the inequality does not change.
1. **Divide both sides by H**:
\[
\frac{Hy}{H} < \frac{7}{H}
\]
This simplifies to:
\[
y < \frac{7}{H}
\]
### Conclusion:
The correct way to represent the solution is:
**A). \( y < \frac{7}{H} \)**.
Answered by
GPT-4o mini
In the inequality \( Bx + 72 < 12 \), we need to identify the term that contains a letter coefficient.
A term is considered a "letter coefficient" if it is a term that includes a variable multiplied by a constant. In this case:
- \( Bx \) is the term where \( B \) is the coefficient (which is a letter because it represents a constant value) and \( x \) is the variable.
- \( 72 \) is a constant term.
- \( 12 \) is also a constant term.
Therefore, the term that is a letter coefficient in the inequality \( Bx + 72 < 12 \) is:
**\( Bx \)**, where \( B \) is the letter coefficient.
A term is considered a "letter coefficient" if it is a term that includes a variable multiplied by a constant. In this case:
- \( Bx \) is the term where \( B \) is the coefficient (which is a letter because it represents a constant value) and \( x \) is the variable.
- \( 72 \) is a constant term.
- \( 12 \) is also a constant term.
Therefore, the term that is a letter coefficient in the inequality \( Bx + 72 < 12 \) is:
**\( Bx \)**, where \( B \) is the letter coefficient.
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