SAT Linear Inequalities Question 79 | Free SAT Prep | Aniko

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Question 79·Easy·Linear Inequalities in One or Two Variables

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A bakery sells cupcakes for $3 each and charges a flat decoration fee of $8 per order. Ella has at most $50 to spend. If she buys only whole cupcakes, what is the greatest number of cupcakes she can order?

Your answer

(Express the answer as an integer)

For inequality word problems, start by defining a variable and writing a total-cost or total-amount expression. Translate key phrases like "at most" into symbols ("at most" means $\le$ ) and set up an inequality. Solve it like a regular equation (undo additions/subtractions, then multiplications/divisions), then interpret the result in context: if the quantity must be whole (people, items), pick the largest or smallest integer that satisfies the inequality as the question asks.

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Write an expression for the total cost

Let $c$ be the number of cupcakes. Each cupcake is $3$ dollars, and there is a flat $8$ -dollar decoration fee. How can you write the total cost in terms of $c$ ?

Turn the words into an inequality

The phrase "at most $50" tells you something about how the total cost compares to $50. Should you use $<$ , $>$ , $\le$ , or $\ge$ ?

Solve for the number of cupcakes

Once you have an inequality with $c$ , solve it step by step (undo the $+8$ , then undo the $\times 3$ ). Remember she can only buy whole cupcakes and you want the greatest number that still fits the inequality.

Desmos Guide

Use Desmos to compute the maximum possible cupcakes

In the expression line, type (50 - 8) / 3 and press Enter. This calculates the largest value of $c$ that makes $3c + 8$ equal to 50. Since Ella must buy whole cupcakes and the total cost must be at most $50, take the greatest whole number less than or equal to the value Desmos shows.

Step-by-step Explanation

Define a variable and write the cost expression

Let $c$ be the number of cupcakes Ella buys.

Each cupcake costs $3$ , so $c$ cupcakes cost $3c$ dollars.

There is also a flat decoration fee of $8$ dollars per order, so the total cost is:

\text{Total cost} = 3c + 8

Translate "at most $50" into an inequality

"At most $50" means the total cost can be less than or equal to $50.

So we write the inequality:

3c + 8 \le 50

Solve the inequality for the number of cupcakes

We solve $3c + 8 \le 50$ step by step.

First subtract $8$ from both sides:

\begin{aligned} 3c + 8 - 8 &\le 50 - 8 \\ 3c &\le 42 \end{aligned}

Now divide both sides by $3$ :

\begin{aligned} \frac{3c}{3} &\le \frac{42}{3} \\ c &\le 14 \end{aligned}

This tells us that Ella can buy any number of cupcakes less than or equal to $14$ , as long as it is a whole number.

Interpret the result in the context of the problem

Cupcakes must be bought as whole numbers, and $c \le 14$ .

The greatest whole number that satisfies $c \le 14$ is $14$ .

Check the cost:

3(14) + 8 = 42 + 8 = 50

This is within her budget of at most $50, so the greatest number of cupcakes she can order is 14.

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation