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Monica is a baby sitter for 2 neighbors.

What we know is:
-She cannot work for any more than 10 hours.
-She charges one family $12 per hour for 2 kids.
-She charges the other family $15 per hour for 3 kids.
-At least $100 is needed for the week for car payment.

X will represent the first family, while Y will represent the second. Make an inequality from this.

7 answers

Let \( x \) represent the number of hours Monica works for the first family (Family X), and \( y \) represent the number of hours she works for the second family (Family Y).

Based on the information provided, we can summarize the scenario with the following inequalities:

  1. Total hours worked should not exceed 10 hours: \[ x + y \leq 10 \]

  2. Total earnings must be at least $100:

    • The earnings from Family X for \( x \) hours are \( 12x \).
    • The earnings from Family Y for \( y \) hours are \( 15y \). Thus, the total earnings inequality is: \[ 12x + 15y \geq 100 \]

Putting these two inequalities together, we have:

\[ \begin{cases} x + y \leq 10 \ 12x + 15y \geq 100 \end{cases} \]

These inequalities can be used to determine how many hours Monica can work for each family to meet the conditions specified.

The format is:

_+_ ≤ __

Got it! Let's express the inequalities in the specified format.

  1. Total hours worked:

    \[ x + y \leq 10 \]

  2. Total earnings:

    Since we want this in the format of a single inequality, we can rewrite the second inequality \( 12x + 15y \geq 100 \) as:

    \[ -12x - 15y \leq -100 \]

    However, typically when converting to the desired format, we express it in terms of non-negative values, so we can keep it as two separate inequalities for clarity.

Finally, combining them in their individual formats for the requirements we have:

  1. \( x + y \leq 10 \)
  2. \( -12x - 15y \leq -100 \) (keeping the negative format here)

If both inequalities need to be presented together in a single line format, you would express it as:

\[ x + y \leq 10 \quad \text{and} \quad -12x - 15y \leq -100 \]

However, to stick closely to the request of one inequality in the format _ + _ ≤ __, here is your first inequality:

\[ x + y \leq 10 \]

Please let me know if you need any further modifications!

Now, the format is:

___ + ___ ≤ ___

Got it! Let's express the inequalities in the desired format:

  1. Total hours worked:

    This is already in the correct format: \[ x + y \leq 10 \]

  2. Total earnings:

    To reformat the earnings condition \(12x + 15y \geq 100\) into the requested style \( ___ + ___ \leq ___ \), we can reorganize it as follows:

    \[ 12x + 15y \geq 100 \implies -12x - 15y \leq -100 \]

This can now be structured in the same format, even though we're keeping it as a negative expression. It still meets the request:

The two inequalities in the desired format are:

  1. \( x + y \leq 10 \)
  2. \( -12x - 15y \leq -100 \)

If you want both inequalities presented explicitly in the format of \( ___ + ___ \leq ___ \), we represent them as follows:

  1. For hours: \[ x + y \leq 10 \]

  2. For earnings (converting inequality): \[ 12x + 15y \geq 100 \quad \text{which is not required explicitly in your desired format since it’s a "≥" inequality.} \] To match, we focus on the other format: \[ 12x + 15y \leq 100 \quad (\text{negating the inequality needed re-expression}) \]

Ultimately, in the exact ___ + ___ ≤ ___ format, you would maintain:

  1. Hours: \[ x + y \leq 10 \]

And we can summarize:

  • For earnings directly in desired "less than or equal to" is not directly convertible in the same method while keeping the conditions sequential to the restrictions.

Thus, please confirm how to represent earnings purposes when only needing hours concretely outlined!

What combinations can be done?

Reminder / Requirements:
-At least $100 needed
-No more than 10 hours worth of work

a. 4 hours for the first neighbor and 5 for the second
b. 7 hours for the first
c. 2 hours for the first and 5 for the second
d. 5 hours for the first and 2 hours for the second

To determine which combinations can meet the requirements, we need to evaluate each option based on two criteria:

  1. Total hours worked must not exceed 10.
  2. Total earnings must be at least $100.

Let's analyze each of the proposed combinations:

a. 4 hours for the first neighbor and 5 for the second

  • Hours Check: \[ 4 + 5 = 9 \quad (\text{which is } \leq 10) \]
  • Earnings Check: \[ (12 \times 4) + (15 \times 5) = 48 + 75 = 123 \quad (\text{which is } \geq 100) \]

Conclusion for a: This combination works.

b. 7 hours for the first neighbor

  • Hours Check: \[ 7 + 0 = 7 \quad (\text{which is } \leq 10) \]
  • Earnings Check: \[ 12 \times 7 = 84 \quad (\text{which is } < 100) \]

Conclusion for b: This combination does not work.

c. 2 hours for the first and 5 for the second

  • Hours Check: \[ 2 + 5 = 7 \quad (\text{which is } \leq 10) \]
  • Earnings Check: \[ (12 \times 2) + (15 \times 5) = 24 + 75 = 99 \quad (\text{which is } < 100) \]

Conclusion for c: This combination does not work.

d. 5 hours for the first and 2 hours for the second

  • Hours Check: \[ 5 + 2 = 7 \quad (\text{which is } \leq 10) \]
  • Earnings Check: \[ (12 \times 5) + (15 \times 2) = 60 + 30 = 90 \quad (\text{which is } < 100) \]

Conclusion for d: This combination does not work.

Summary of Results:

  • Combination a (4 hours for the first neighbor and 5 for the second): Works.
  • Combination b (7 hours for the first): Does not work.
  • Combination c (2 hours for the first and 5 for the second): Does not work.
  • Combination d (5 hours for the first and 2 for the second): Does not work.

The only valid combination based on the requirements is option a: 4 hours for the first family and 5 hours for the second family.