Question

A salad dressing recipe requires at least 6 oz of oil to be combined with some combination of vinegar and lemon juice in a 16 oz container.

What inequality models this situation?

Let x represent the number of ounces of vinegar and let y represent the number of ounces of lemon juice.

Answer 1

A company prepares a graph of its profit forecast. The annual budget predicts monthly income of at least $5000 more than expenditures.

What inequality models this situation?

Let x represent the expenditures in dollars for a given month and let y represent the income in dollars for the same month.

Express your answer as an inequality in slope-intercept form.

Answer 2

To model the salad dressing recipe's requirements, let's summarize the information given:

1. The total volume of the salad dressing (oil, vinegar, and lemon juice combined) should not exceed 16 ounces.
2. There must be at least 6 ounces of oil in the dressing.

Let:
- \( x \) = ounces of vinegar
- \( y \) = ounces of lemon juice
- The amount of oil = 6 ounces

Since we are using a total of 16 ounces for the container, the combined volume of the oil, vinegar, and lemon juice can be represented as follows:

\[
6 + x + y \leq 16
\]

Subtracting 6 from both sides, we can rewrite this inequality as:

\[
x + y \leq 10
\]

Now, since there are no restrictions on how much vinegar and lemon juice can be used individually (except for their combined total), we also have the conditions that both \( x \) and \( y \) should be non-negative:

\[
x \geq 0
\]
\[
y \geq 0
\]

Putting it all together, the inequalities that model this situation are:

1. \( x + y \leq 10 \)
2. \( x \geq 0 \)
3. \( y \geq 0 \)

These inequalities appropriately represent the constraints of the salad dressing recipe involving oil, vinegar, and lemon juice.

Answer 3

To model the situation where the monthly income is at least $5000 more than the expenditures, we start by defining the variables:

- Let \( x \) represent the expenditures (in dollars).
- Let \( y \) represent the income (in dollars).

The condition states that the income \( y \) is at least $5000 more than the expenditures \( x \). We can express this condition with the following inequality:

\[
y \geq x + 5000
\]

This inequality shows that the income must be greater than or equal to the expenditures plus $5000.

To express this inequality in slope-intercept form, we can rewrite it as:

\[
y - x \geq 5000
\]

This can also be rearranged to match the traditional slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. To do this, we simply shift \( x \) to the other side of the inequality:

\[
y \geq x + 5000
\]

In summary, the correct inequality that models the situation in slope-intercept form is:

\[
y \geq x + 5000
\]

This inequality indicates that for any given expenditure \( x \), the income \( y \) must be at least $5000 greater than that expenditure.