A 10-gallon water cooler in an office provides water for the whole department. Each hour, 30 ounces of water are removed from the cooler and drunk by office workers. Write an equation to show how long the water in the cooler will last. (10 gallons is 1,280 ounces.)

Question

Responses:

1,280−30h=0

10−30h=0

1,280+30h=0

30h=10

Answer 1

To write an equation that shows how long the water in the cooler will last, we know that the total amount of water in the cooler is 1,280 ounces (10 gallons), and every hour, 30 ounces are taken out.

We can set up the equation based on the relationship between the initial amount of water and the amount of water removed:

1,280 - 30h = 0

This equation states that when the initial amount of water (1,280 ounces) minus the amount of water removed after h hours (30 ounces per hour times h hours) equals zero, then the water has been fully consumed.

Therefore, the correct response is:

1,280 - 30h = 0

Answer 2

Justify your steps using the properties of equality and determine the number of solutions to the equation 3/2(4−8x)+5x=1−7x

Answer 3

To solve the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \), we will simplify and justify each step using the properties of equality.

Step 1: Distribute \( \frac{3}{2} \) on the left side

We use the distributive property:

\[ \frac{3}{2} \cdot 4 - \frac{3}{2} \cdot 8x + 5x = 1 - 7x \]

Calculating \( \frac{3}{2} \cdot 4 \) gives \( 6 \), and \( -\frac{3}{2} \cdot 8x \) gives \( -12x \). Therefore, we have:

\[ 6 - 12x + 5x = 1 - 7x \]

Step 2: Combine like terms on the left side

Combine \( -12x + 5x \):

\[ 6 - 7x = 1 - 7x \]

Step 3: Add \( 7x \) to both sides

By using the addition property of equality, we can eliminate \( -7x \) from both sides:

\[ 6 - 7x + 7x = 1 - 7x + 7x \]

This simplifies to:

\[ 6 = 1 \]

Step 4: Analyze the result

The equation \( 6 = 1 \) is a contradiction, meaning it is never true.

Conclusion

Since we reached a statement that is not true, this indicates that the original equation has no solutions. Thus, the number of solutions to the equation \( \frac{3}{2}(4 - 8x) + 5x = 1 - 7x \) is:

0 solutions.

Answer 4

Using the properties of inequality, justify both steps in solving the inequality 2/5m−7≤17

Answer 5

To solve the inequality \( \frac{2}{5}m - 7 \leq 17 \), we will justify each step using properties of inequality.

Step 1: Add 7 to both sides

To isolate the term involving \( m \), we’ll add 7 to both sides of the inequality. This is justified by the Addition Property of Inequalities, which states that if \( a \leq b \), then \( a + c \leq b + c \) for any number \( c \).

\[ \frac{2}{5}m - 7 + 7 \leq 17 + 7 \]

This simplifies to:

\[ \frac{2}{5}m \leq 24 \]

Step 2: Multiply both sides by \( \frac{5}{2} \)

Next, we want to eliminate the coefficient of \( m \) by multiplying both sides by \( \frac{5}{2} \). This is justified by the Multiplication Property of Inequalities, which states that if \( a \leq b \) and \( c > 0 \), then \( ac \leq bc \). In this case, \( c \) is \( \frac{5}{2} \), which is positive.

\[ \frac{5}{2} \cdot \frac{2}{5}m \leq 24 \cdot \frac{5}{2} \]

The left side simplifies to \( m \):

\[ m \leq 24 \cdot \frac{5}{2} \]

Calculating the right side:

\[ 24 \cdot \frac{5}{2} = \frac{120}{2} = 60 \]

Conclusion

So, the solution to the inequality is:

\[ m \leq 60 \]

Therefore, using the properties of inequality, we justified each step leading to the final solution.

Answer 6

Rewrite the equation 4x3x−12+3=10
in two equations joined by “and.”

Answer 7

To rewrite the equation \( 4x + 3x - 12 + 3 = 10 \) as two equations joined by "and," we first need to simplify the given equation.

Step 1: Combine like terms

Combine the terms on the left side:

\[ (4x + 3x) + (-12 + 3) = 10 \]

This becomes:

\[ 7x - 9 = 10 \]

Step 2: Rewrite as two separate equations

Now that we have a simplified equation, we can separate it into two equations joined by "and." One equation can represent the linear part, while the other can represent the constant part:

1. \( 7x - 9 = 10 \)
2. \( -9 = 10 \) (This is not true but is part of the separation process)

However, typically, you may want to separate the expression into parts that maintain the equality but form two valid equations. Here’s a correct interpretation:

• \( 7x = 10 + 9 \)
• \( 10 + 9 = 19 \)

So we can rewrite the original equation as:

1. \( 7x = 19 \)
2. \( 10 = 19 \) (which is a contradiction, giving insight that the equations can’t hold both)

Final Summary

Thus, a clearer separation following the original logic would be:

Final Answer:

• \( 7x - 9 = 10 \) and \( x = \frac{19}{7} \)

This gives us two equations representing the splitting of the terms.

Answer 8

Rearrange the slope formula y=mx+b
for the quantity of interest m
. Make sure to show each step.

Answer 9

To rearrange the slope-intercept formula \( y = mx + b \) for \( m \), we will isolate \( m \) on one side. Here are the steps:

Step 1: Subtract \( b \) from both sides

We start with the original equation:

\[ y = mx + b \]

To isolate the term containing \( m \), we first subtract \( b \) from both sides. This is justified by the Subtraction Property of Equality, which states that if \( a = b \), then \( a - c = b - c \) for any number \( c \).

\[ y - b = mx \]

Step 2: Divide both sides by \( x \)

Next, we want to isolate \( m \). Since \( mx \) means \( m \) multiplied by \( x \), we can divide both sides by \( x \) (assuming \( x \neq 0 \)). This is justified by the Division Property of Equality, which states that if \( a = b \) and \( c \neq 0 \), then \( \frac{a}{c} = \frac{b}{c} \).

\[ \frac{y - b}{x} = m \]

Final Result

Now we have \( m \) isolated:

\[ m = \frac{y - b}{x} \]

Thus, the slope \( m \) can be expressed in terms of \( y \), \( b \), and \( x \):

Final Answer:

\[ m = \frac{y - b}{x} \]

Answer 10

ezeree received a $100 gift card and needs new sweaters for school. The sweaters cost $27.50 each. Write an equation to determine how many sweaters she can buy and spend exactly $100, solve it, and determine if the value is a viable solution.

Answer 11

To determine how many sweaters Ezeree can buy with a $100 gift card, we can set up an equation based on the total cost of the sweaters.

Step 1: Define the variables

Let \( x \) be the number of sweaters Ezeree can buy. Each sweater costs $27.50.

Step 2: Write the equation

The total cost of \( x \) sweaters can be expressed as:

\[ 27.50x \]

Since Ezeree wants to spend exactly $100, we can set up the equation:

\[ 27.50x = 100 \]

Step 3: Solve for \( x \)

To find \( x \), we'll divide both sides of the equation by 27.50:

\[ x = \frac{100}{27.50} \]

Calculating this:

\[ x = 3.63636\ldots \]

Since \( x \) must be a whole number (you can’t buy a fraction of a sweater), we round down to the nearest whole number, which is \( 3 \).

Step 4: Determine the total cost for 3 sweaters

Next, we can check if buying 3 sweaters will allow her to spend exactly $100. The cost for 3 sweaters is:

\[ 27.50 \times 3 = 82.50 \]

Step 5: Checking for 4 sweaters

To see if she could spend up to $100, we can check the cost for 4 sweaters:

\[ 27.50 \times 4 = 110 \]

Conclusion

• If Ezeree buys 3 sweaters, she spends $82.50, which does not use the full value of the gift card.
• If she buys 4 sweaters, she would spend $110, which exceeds the gift card amount.

Thus, the only viable solution is:

Ezeree can buy 3 sweaters.