SAT Linear Inequalities Question 80 | Free SAT Prep | Aniko

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Question 80·Medium·Linear Inequalities in One or Two Variables

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A rideshare service charges a total cost $T$ (in dollars) according to the model $T = 1.75d + 2.50w + 3$ , where $d$ is the number of miles traveled and $w$ is the number of minutes spent waiting. During a promotion, rides are limited to $4 \le d \le 8$ and $0 \le w \le 10$ .

Which of the following inequalities gives all possible values of $T$ for a promotional ride?

When you see a linear expression like $T = ax + by + c$ with positive coefficients and given ranges for $x$ and $y$ , you can quickly find the range of $T$ by plugging in the extreme values: use the smallest allowed values of the variables to get the minimum, and the largest values to get the maximum. Be careful to include any constant term (like the +3 here) in both calculations and to respect any lower bounds on the variables (such as $d \ge 4$ ). Then match these two endpoint values to the appropriate inequality in the choices.

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Look at how T depends on d and w

In the formula $T = 1.75d + 2.50w + 3$ , notice that both $1.75$ and $2.50$ are positive. What happens to $T$ when $d$ or $w$ gets larger?

Use the endpoints of the ranges

To find the smallest and largest possible values of $T$ , think about which combinations of $d$ and $w$ (using their allowed ranges) would make $T$ as small as possible and as large as possible.

Substitute extreme values into the formula

Try plugging the smallest allowed $d$ and $w$ into $T = 1.75d + 2.50w + 3$ , then plug the largest allowed $d$ and $w$ into the same formula. These two results will form the endpoints of your inequality for $T$ .

Match your endpoints to an answer choice

Once you have the smallest and largest $T$ from your substitutions, look at the answer choices and find the inequality whose lower and upper bounds match the values you found.

Desmos Guide

Compute the minimum possible T

In a Desmos expression line, type 1.75*4 + 2.50*0 + 3 to evaluate the cost when $d = 4$ and $w = 0$ . Note the numerical result; this is the smallest possible value of $T$ .

Compute the maximum possible T

In another expression line, type 1.75*8 + 2.50*10 + 3 to evaluate the cost when $d = 8$ and $w = 10$ . Note this result; it is the largest possible value of $T$ .

Match to an answer choice

Look at the two values you found in Desmos (the minimum and maximum). Find the answer choice whose inequality has these two numbers as its lower and upper bounds.

Step-by-step Explanation

Understand the cost model and the constraints

The total cost is given by $T = 1.75d + 2.50w + 3$ , where:

$d$ is miles traveled
$w$ is minutes waiting

You are told that during the promotion:

$4 \le d \le 8$
$0 \le w \le 10$

Both coefficients, $1.75$ and $2.50$ , are positive, so increasing $d$ or $w$ will increase $T$ .

Find the smallest possible cost

To get the smallest $T$ , use the smallest allowed values of $d$ and $w$ :

Smallest $d$ is $4$
Smallest $w$ is $0$

Substitute $d = 4$ and $w = 0$ into the formula:

T = 1.75(4) + 2.50(0) + 3

Now simplify this expression to get the minimum value of $T$ .

Compute the minimum cost

Continue simplifying:

T = 1.75(4) + 2.50(0) + 3 \\[0.7em] T = 7 + 0 + 3 \\[0.7em] T = 10

So the smallest possible total cost is $T = 10$ .

Find the largest possible cost

To get the largest $T$ , use the largest allowed values of $d$ and $w$ :

Largest $d$ is $8$
Largest $w$ is $10$

Substitute $d = 8$ and $w = 10$ into the formula:

T = 1.75(8) + 2.50(10) + 3

Now simplify this expression to get the maximum value of $T$ .

Compute the maximum cost and write the inequality

Simplify the expression for the maximum cost:

T = 1.75(8) + 2.50(10) + 3 \\[0.7em] T = 14 + 25 + 3 \\[0.7em] T = 42

So $T$ can be as small as $10$ and as large as $42$ . This means $T$ must satisfy the inequality

10 \le T \le 42

Among the answer choices, this corresponds to choice A, $10 \le T \le 42$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation