tom is deciding whether or not he should become a member of the gym to use their basketball courts the membership cost is $125. members pay $3 to rent out the basketball courts how many times can tom play basketball if he wants to spend no more than $200

Question

this includes a numberline

Answer 1

To determine how many times Tom can play basketball while keeping his total spending at or below $200, we can set up the following equation.

1. Membership Cost: $125 (one-time fee)
2. Court Rental Fee for Members: $3 per use
3. Total Budget: $200

Step 1: Calculate the remaining budget after the membership fee.

Remaining budget = Total budget - Membership cost
Remaining budget = $200 - $125 = $75

Step 2: Determine how many times Tom can rent the basketball court.

Let \( x \) be the number of times Tom plays basketball.

The cost for renting the court \( x \) times = \( 3x \).

We need to satisfy the following inequality:

\[ 3x \leq 75 \]

Step 3: Solve for \( x \).

Dividing both sides of the inequality by 3:

\[ x \leq \frac{75}{3} \]

\[ x \leq 25 \]

So, Tom can play basketball a maximum of 25 times while keeping his expenses at or below $200.

Summary

• Membership cost: $125
• Remaining budget for court rental: $75
• Cost per rental: $3
• Maximum number of rentals (times Tom can play basketball): 25

Number Line Illustrating Tom's Budget

You can visualize a number line to represent Tom's spending:

|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|
0    25   50   75   100  125  150  175  200
             |----------------|
                  $200 (Total Budget)
                  |----------|
                $125 (Membership Cost)

• The left part of the line represents the total money spent, reaching from $0 to $200.
• The segment from $125 to $200 indicates how much of his budget is left after paying for the membership.
• The maximum number of rentals \( x \) possible (up to $75) shows he can play basketball \( x \) times without exceeding the budget.

Thus, Tom can play basketball up to 25 times for the total cost to remain within his budget of $200.

Answer 2

5x + ( x - 8 ) > 9/2 ( 2x - 4 )

Answer 3

Let's solve the inequality step by step:

Given:

\[ 5x + (x - 8) > \frac{9}{2}(2x - 4) \]

Step 1: Simplify both sides of the inequality.

First, simplify the left side:

\[ 5x + x - 8 = 6x - 8 \]

Now simplify the right side:

\[ \frac{9}{2}(2x - 4) = \frac{9}{2} \cdot 2x - \frac{9}{2} \cdot 4 = 9x - 18 \]

So, the inequality simplifies to:

\[ 6x - 8 > 9x - 18 \]

Step 2: Rearrange the inequality.

To isolate \( x \), first move \( 6x \) to the right side and move \( -18 \) to the left side:

\[ -8 + 18 > 9x - 6x \]

This simplifies to:

\[ 10 > 3x \]

Step 3: Solve for \( x \).

Dividing both sides by 3:

\[ \frac{10}{3} > x \]

This can also be written as:

\[ x < \frac{10}{3} \]

Step 4: Interval Notation (optional).

In interval notation, the solution set can be expressed as:

\[ (-\infty, \frac{10}{3}) \]

Final Answer

Thus, the solution to the inequality is:

\[ x < \frac{10}{3} \]

Answer 4

this one also includes a number line so to the left or right would you say?

Answer 5

To illustrate the solution \( x < \frac{10}{3} \) on a number line, we'll place the value \( \frac{10}{3} \) (approximately \( 3.33 \)) on the number line and shade to the left of it, since the solution indicates that \( x \) can take any value less than \( \frac{10}{3} \).

Number Line Illustration

Here’s how the number line would look:

|----|----|----|----|----|----|----|----|----|----|----|
-3   -2   -1    0    1    2    3   3.33   4    5
              |<-------------------|
              -∞               10/3

Key Points:

• The point \( \frac{10}{3} \) is marked on the number line.
• An open circle (or just not filled) at \( \frac{10}{3} \) indicates that this value is not included in the solution (since the inequality is strict, \( < \)).
• The shading to the left signifies that all values less than \( \frac{10}{3} \) are included in the solution.

Summary

So to summarize, the solution \( x < \frac{10}{3} \) on a number line is represented by marking \( \frac{10}{3} \) with an open circle and shading the area to the left, indicating all values lower than \( \frac{10}{3} \) are valid solutions.

Answer 6

what does leq mean on the first question I asked

Answer 7

In mathematical notation, "leq" stands for "less than or equal to." It is often written as \( \leq \).

• If you see \( x \leq a \), it means that \( x \) can be any value less than \( a \) or equal to \( a \).

For example, if you have \( x \leq 5 \), the possible values of \( x \) include all numbers less than 5 and the number 5 itself.

If you were referring to a specific inequality in your earlier question that included the term "leq," please let me know, and I can clarify further!

Answer 8

a teacher is choosing between two options for a class field trip to an orchard
at orchard a admission cots $9 per person and 3 caperones are required
at orchard b the cost is $10 per person but only 1 chaperone is required
at each orchard the same price applies to both chaperones and students

to help her compare the cost of her two options the teacher first writes the equation 9(n+3) = 10(n+1) and then she writes the inequality 9(n+3)<10(n+1)

Answer 9

Let's start by analyzing both the equation and the inequality that the teacher wrote.

Understanding the Cost Equation

The teacher is comparing costs for a field trip to two orchards:

1. Orchard A:

   • Admission cost: $9 per person
   • Chaperones required: 3
   • Total cost formula: \( 9(n + 3) \)
   • Here, \( n \) represents the number of students.

2. Orchard B:

   • Admission cost: $10 per person
   • Chaperones required: 1
   • Total cost formula: \( 10(n + 1) \)

The equation \( 9(n + 3) = 10(n + 1) \) represents the situation where the costs of both orchards are equal. To find the value of \( n \) (the number of students) where the costs are the same, we can solve the equation.

Solving the Equation

\[ 9(n + 3) = 10(n + 1) \]

Distributing both sides:

\[ 9n + 27 = 10n + 10 \]

Now, rearranging to isolate \( n \):

1. Move \( 9n \) to the right:

\[ 27 = 10n - 9n + 10 \]

2. This simplifies to:

\[ 27 = n + 10 \]

3. Finally, subtract 10 from both sides:

\[ n = 27 - 10 \]

\[ n = 17 \]

Thus, the costs of both orchards are equal when there are 17 students.

Understanding the Inequality

The inequality \( 9(n + 3) < 10(n + 1) \) means that we are comparing the costs and trying to find when the cost of Orchard A is less than the cost of Orchard B:

Solving the Inequality

We solve it similarly to the equation:

1. Start with the inequality:

\[ 9(n + 3) < 10(n + 1) \]

2. Distributing both sides:

\[ 9n + 27 < 10n + 10 \]

3. Rearranging to isolate \( n \):

\[ 27 - 10 < 10n - 9n \]

\[ 17 < n \]

This can be rewritten as:

\[ n > 17 \]

Conclusion

• The equation \( 9(n + 3) = 10(n + 1) \) shows that the costs are equal when there are 17 students.
• The inequality \( 9(n + 3) < 10(n + 1) \) shows that the cost of Orchard A is less than the cost of Orchard B when there are more than 17 students.

Visualizing with Number Line:

On a number line:

|----|----|----|----|----|----|----|----|
...   15   16    17   18   19   20   ...
          |
          >------------------- (n > 17)

Here, the point at \( n = 17 \) is marked, with an arrow indicating that all values to the right (greater than 17) satisfy the inequality \( 9(n + 3) < 10(n + 1) \).

Answer 10

what do we know?

Answer 11

Based on the problem you've described regarding the field trip options at two orchards, here's a summary of what we know:

Key Information

1. Orchard A:

   • Admission Cost: $9 per person
   • Chaperones Required: 3
   • Total Cost Formula: \( 9(n + 3) \), where \( n \) is the number of students.

2. Orchard B:

   • Admission Cost: $10 per person
   • Chaperones Required: 1
   • Total Cost Formula: \( 10(n + 1) \)

Equations and Inequalities

• Equation:

  • The teacher wrote the equation \( 9(n + 3) = 10(n + 1) \) to find out how many students \( n \) make both costs equal.
  • Solution to the equation: The costs of the two orchards are equal when there are 17 students.

• Inequality:

  • The teacher wrote the inequality \( 9(n + 3) < 10(n + 1) \) to determine when the cost of Orchard A is less than the cost of Orchard B.
  • Solution to the inequality: The cost of Orchard A is less than Orchard B when there are more than 17 students (i.e., \( n > 17 \)).

What Does This Mean?

• For 17 Students:

  • The total cost for both orchards will be the same.

• For More Than 17 Students:

  • Orchard A becomes the more economical option, as its total cost will be lower than that of Orchard B.

• For Fewer Than 17 Students:

  • Orchard B will be the less expensive option in this case.

Summary of Findings

Based on the information and calculations:

• At 17 students, the costs for both options are equal.
• If the teacher has more than 17 students, she should choose Orchard A to save money.
• If the teacher has fewer than 17 students, she should choose Orchard B to minimize costs.

This analysis helps the teacher make an informed decision about which orchard to choose for the field trip based on the number of students and required chaperones.