Question

buddy’s sugar addiction

1. papa elf determines that an elf’s daily sugar intake must be less than 100 grams to maintain peak christmas cheer. if buddy consumes (2x - 10) grams of sugar, write and solve the inequality to determine how much more sugar he can consume.

graphing equations

1. graph (2x + 3y = 18), representing the number of christmas decorations buddy hangs at the hobbs’ apartment compared to those he wants to hang.

write and solve systems of equations

1. at gimbels department store, buddy needs to stock (x) cotton - headed ninja baby dolls and (y) christmas tree ornaments. cotton - headed ninja baby dolls cost $15 each, and ornaments cost $10 each. if he needs to stock 25 total items and the total cost is $300, write and solve the system to determine how many ninja baby dolls buddy has.
2. buddy tracks his daily activities: (x) minutes spent spreading christmas cheer and (y) minutes spent eating sugar. his total active time is 180 minutes, and cheer - spreading takes 30 more minutes than sugar consumption. write and solve the system to determine the number of minutes he spends eating sugar.

spaghetti syrup function!

1. buddy’s famous spaghetti and syrup recipe follows a unique mathematical function. the amount of syrup (in tablespoons) can be calculated by the function (f(x)=4x^{2}+3x + 10), where (x) represents the number of marshmallows added to the dish.

if buddy adds 2 marshmallows to his spaghetti, how many tablespoons of syrup will he use?

Explanation:

Problem 6: Buddy’s Sugar Addiction

Step1: Define the inequality

Let \( x \) be the additional sugar (in grams) Buddy can consume. His current intake is \( 2x - 10 \) grams, and total intake must be \( < 100 \) grams. So, the inequality is \( (2x - 10) + x < 100 \)? Wait, no—wait, the problem says "consumes \( 2x - 10 \) grams" and we need to find how much more (\( x \)) he can consume so that total is \( < 100 \). Wait, maybe re-express: Let \( y \) be total sugar, \( y = (2x - 10) + \text{additional} \), but maybe the problem means his current consumption is \( 2x - 10 \), and total must be \( < 100 \). Wait, the problem says: "write and solve the inequality to determine how much more sugar he can consume." Let \( x \) be the additional sugar. Then total sugar is \( (2x - 10) + x \)? No, maybe the current consumption is \( 2x - 10 \), and he can consume \( x \) more, so total \( (2x - 10) + x < 100 \)? Wait, no—maybe the variable is miswritten. Wait, the problem says "consumes \( 2x - 10 \) grams"—maybe that's a typo, or \( x \) is the additional. Wait, let's re-express: Let \( s \) be the additional sugar. Then total sugar is \( (2x - 10) + s < 100 \)? No, the problem is likely: His current sugar intake is \( 2x - 10 \), and he can consume \( x \) more, so total \( (2x - 10) + x < 100 \)? Wait, no—maybe the inequality is \( (2x - 10) + x < 100 \), but that seems off. Wait, maybe the current intake is \( 2x - 10 \), and total must be \( < 100 \), so \( 2x - 10 + x < 100 \)? No, perhaps the problem is: Let \( x \) be the additional sugar. Then total sugar is \( (2x - 10) + x < 100 \)? Wait, maybe I misread. Let's start over:

"an elf’s daily sugar intake must be less than 100 grams. If Buddy consumes \( 2x - 10 \) grams of sugar, write and solve the inequality to determine how much more sugar he can consume."

Let \( y \) be the additional sugar (grams). Then total sugar is \( (2x - 10) + y < 100 \). But the problem uses \( x \) for additional? Wait, maybe the variable is \( x \) for additional. So total sugar: \( (2x - 10) + x < 100 \)? No, that doesn't make sense. Wait, maybe the current consumption is \( 2x - 10 \), and he can consume \( x \) more, so total \( (2x - 10) + x < 100 \)? Wait, no—maybe the inequality is \( (2x - 10) + x < 100 \), but solving: \( 3x - 10 < 100 \) → \( 3x < 110 \) → \( x < \frac{110}{3} \approx 36.67 \). But this seems confusing. Alternatively, maybe the current intake is \( 2x - 10 \), and total must be \( < 100 \), so \( 2x - 10 < 100 \)? But that would be solving \( 2x - 10 < 100 \) → \( 2x < 110 \) → \( x < 55 \). Wait, the problem says "how much more"—maybe the current intake is \( 2x - 10 \), and he can consume \( x \) more, so total \( (2x - 10) + x < 100 \), but that's unclear. Maybe the problem has a typo, but assuming the inequality is \( 2x - 10 < 100 \) (treating \( x \) as total additional), then:

Step1: Write the inequality

\( 2x - 10 < 100 \)

Step2: Solve for \( x \)

Add 10 to both sides: \( 2x < 110 \)
Divide by 2: \( x < 55 \)

Problem 7: Graph \( 2x + 3y = 18 \)

Step1: Find intercepts

• For \( x \)-intercept: Set \( y = 0 \), so \( 2x = 18 \) → \( x = 9 \). Point: \( (9, 0) \).
• For \( y \)-intercept: Set \( x = 0 \), so \( 3y = 18 \) → \( y = 6 \). Point: \( (0, 6) \).

Step2: Plot points and draw line

Plot \( (9, 0) \) and \( (0, 6) \) on the grid, then draw a straight line through them.

Problem 8: Systems of Equations (Gimbels Stocking)

Step1: Define variables and equations

Let \( x \) = ninja baby dolls, \( y \) = ornaments.

• Total items: \( x + y = 25 \)
• Total cost: \( 15x + 10y = 300 \)

Step2: Solve the system

From first equation: \( y = 25 - x \). Substitute into second equation:
\( 15x + 10(25 - x) = 300 \)
\( 15x + 250 - 10x = 300 \)
\( 5x + 250 = 300 \)
\( 5x = 50 \)
\( x = 10 \)

Problem 9: Systems of Equations (Daily Activities)

Answer:

Step1: Substitute \( x = 2 \) into \( f(x) = 4x^2 + 3x + 10 \)

\( f(2) = 4(2)^2 + 3(2) + 10 \)

Step2: Calculate

\( 4(4) + 6 + 10 = 16 + 6 + 10 = 32 \)

Final Answers (by Problem):

1. \( x < 55 \) (grams)
2. Line through \( (9, 0) \) and \( (0, 6) \)
3. \( x = 10 \) (ninja baby dolls)
4. \( y = 75 \) (minutes eating sugar)
5. \( 32 \) (tablespoons of syrup)