At the beginning of the 2016 season (his 21st season), kicker Adam Vinatieri had 2253 total points.

Let x be the number of successful point-after tries and let y be the number of field goals.
Write an inequality that shows how many successful point-after tries (1 point each) and/or field goals (3 points each) Vinatieri could kick in the 2016 season to surpass Andersen’s record of 2544 points. Convert the inequality to standard form. Enter your answers in the boxes.

___x + ___ y + ___ > ___
x + ___ y > ___

11 answers

To set up the inequality for the situation, we need to account for the points scored from successful point-after tries (PATs) and field goals:

  • Each successful point-after try contributes 1 point (let \( x \) be the number of successful PATs).
  • Each successful field goal contributes 3 points (let \( y \) be the number of field goals).

Given that Adam Vinatieri had 2253 total points at the beginning of the 2016 season, we want to determine how many successful PATs and field goals he could achieve in order to surpass Morten Andersen's record of 2544 total points.

The inequality representing the requirement to surpass Andersen's record is:

\[ 2253 + 1x + 3y > 2544 \]

Subtracting 2253 from both sides gives:

\[ 1x + 3y > 291 \]

Now, we can express this in the standard form \( Ax + By + C > 0 \):

\[ x + 3y - 291 > 0 \]

This can also be written as:

\[ x + 3y + (-291) > 0 \]

To make it clearer, we can represent the standard form as follows:

\[ 1x + 3y - 291 > 0 \]

Thus, the required components of the inequality in standard form are:

  • A: 1 (coefficient of \( x \))
  • B: 3 (coefficient of \( y \))
  • C: -291
  • In the form \( x + By > C \), we have:

\[ x + 3y > 291 \]

So the final answers are:

  • ___x = 1
  • ___y = 3
  • ___C = -291

And for the simplified form:

  • \( x + ___y = 291 \); thus, this is \( x + 3y > 291 \).

1, 3, and -291 for the respective boxes.

Which graph is the graph of the inequality?

A. Coordinate grid with axes labeled x and y. Dashed line passes through ordered pairs 0 comma 97 and 291 comma 0. The area above the line is shaded.

B. Coordinate grid with axes labeled x and y. Dashed line passes through ordered pairs 0 comma 97 and 291 comma 0. The area below the line is shaded.

C. Coordinate grid with axes labeled x and y. Dashed line passes through ordered pairs 97 comma 0 and 0 comma 291. The area above the line is shaded.

D. Coordinate grid with axes labeled x and y. Dashed line passes through ordered pairs 97 comma 0 and 0 comma 291. The area below the line is shaded.

To determine which graph represents the inequality \( x + 3y > 291 \), let's first examine what the inequality can tell us.

  1. Identifying the Boundary Line: The line represented by the equation \( x + 3y = 291 \) will serve as our boundary line. We can find the intercepts to draw this line:

    • x-intercept: Set \( y = 0 \): \[ x + 3(0) = 291 \implies x = 291 \quad \text{(point (291, 0))} \]
    • y-intercept: Set \( x = 0 \): \[ 0 + 3y = 291 \implies y = 97 \quad \text{(point (0, 97))} \]
  2. Graphing the Line: The line will pass through the points \( (0, 97) \) and \( (291, 0) \).

  3. Dashed Line: The inequality \( x + 3y > 291 \) means we cannot include the points on the line itself (the inequality is not "greater than or equal to"), so the line should be dashed.

  4. Shading the Area: Since we have \( x + 3y > 291 \), we need to shade the area above the line, because points above the line will satisfy the inequality.

Now, we can use this information to identify the correct graph among the options provided:

  • Option A: Dashed line through (0, 97) and (291, 0) with the area above shaded. This matches our requirements.
  • Option B: Dashed line through (0, 97) and (291, 0) with the area below shaded. This does not match.
  • Option C: Dashed line through (97, 0) and (0, 291) with the area above shaded. This does not match since the intercepts are incorrect.
  • Option D: Dashed line through (97, 0) and (0, 291) with the area below shaded. This does not match either.

Based on this analysis, the correct graph that represents the inequality \( x + 3y > 291 \) is Option A.

You have to choose one of them.

The correct choice for the graph of the inequality \( x + 3y > 291 \) is:

A.

This option has a dashed line passing through the points \( (0, 97) \) and \( (291, 0) \), with the area above the line shaded, which correctly represents the inequality.

So option B?

No, the correct option is A, not B.

To clarify:

  • Option A has a dashed line through the points \( (0, 97) \) and \( (291, 0) \) with the area above the line shaded, which correctly represents the inequality \( x + 3y > 291 \).

  • Option B also has a dashed line through the same points but with the area below the line shaded, which does not match the inequality.

Thank you for your patience. The correct option is definitely A.

SO OPTION A???

Yes, that's correct! The correct choice is Option A. It features a dashed line through the points \( (0, 97) \) and \( (291, 0) \) with the area above the line shaded, which accurately represents the inequality \( x + 3y > 291 \). Thank you for your understanding!

Ciara budgeted $30 per month for entertainment. Movies cost $7 and sports games cost $5.
Let x represent the number of movies she attended in a given month, and let y represent the number of sports games she attended in the same month.
What inequality models this problem?

To model Ciara's budget for entertainment based on the costs of movies and sports games, we can write an inequality that reflects her total spending.

  1. Budget Limit: Ciara has a budget of $30 per month for entertainment.

  2. Cost of Movies: Each movie costs $7, and if she attends \( x \) movies, the total cost for movies will be \( 7x \).

  3. Cost of Sports Games: Each sports game costs $5, and if she attends \( y \) sports games, the total cost for sports games will be \( 5y \).

  4. Total Cost: The combined cost of attending movies and sports games must not exceed her budget of $30.

Thus, the inequality that represents this situation is:

\[ 7x + 5y \leq 30 \]

In summary, the inequality \( 7x + 5y \leq 30 \) effectively models Ciara's budgeting problem for entertainment.