A mobile phone plan charges a monthly fee of dollars for unlimited texting plus dollars for each minute of voice calls. The user wants their total monthly cost to be at least and at most of the monthly texting fee. Let represent the number of voice‐call minutes the user makes in one month. Which inequality represents all possible values of that satisfy the user’s goal?

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SAT Linear Inequalities Question 135 | Free SAT Prep | Aniko

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Question 135·Hard·Linear Inequalities in One or Two Variables

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A mobile phone plan charges a monthly fee of $p$ dollars for unlimited texting plus $0.08$ dollars for each minute of voice calls. The user wants their total monthly cost to be at least $120\%$ and at most $150\%$ of the monthly texting fee.

Let $v$ represent the number of voice‐call minutes the user makes in one month.

Which inequality represents all possible values of $v$ that satisfy the user’s goal?

For word problems with costs and percentages, first write a clear algebraic expression for the total cost, then translate phrases like "at least 120% and at most 150% of" into a compound inequality using decimal forms (e.g., $1.2p$ and $1.5p$ ). Set the total cost between those two bounds, solve the compound inequality step by step (keeping the units straight: dollars vs. minutes), and only at the end compare your simplified inequality to the answer choices; this avoids common mistakes like dropping a coefficient or mixing up which variable represents which quantity.

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Represent the total cost

What algebraic expression gives the total monthly cost in dollars if the texting fee is $p$ dollars and each voice minute costs $0.08$ dollars and you use $v$ minutes?

Turn the percent condition into numbers

"At least $120\%$ and at most $150\%$ of the monthly texting fee" can be written using decimal multipliers of $p$ . What are $120\%$ and $150\%$ as decimals, and how can you write an inequality with the total cost between those two values?

Solve the compound inequality step by step

After you write an inequality of the form $1.2p \le p + 0.08v \le 1.5p$ , first subtract $p$ from all three parts, then divide everything by $0.08$ . Be careful not to drop the $0.08$ when you solve for $v$ .

Compare your result to the choices

Once you have $v$ between two multiples of $p$ , look for the answer choice where $v$ is bounded below and above by those same multiples of $p$ .

Desmos Guide

Pick a simple value for $p$

In Desmos, define a simple positive value for $p$ , such as typing p = 10 on its own line. This lets you turn the algebra problem into a numerical one while keeping the same relationships.

Enter the total cost function and the bounds

Type C(v) = p + 0.08v to represent the total monthly cost. Then type y = 1.2p and y = 1.5p to draw the lower and upper cost bounds based on the texting fee.

Find the $v$ values where the cost hits each bound

Use the graph or a table: click the gear icon next to C(v) and choose "Table", then look for the $v$ -values where C(v) equals 1.2p and 1.5p. These two $v$ -values are your lower and upper limits for voice-call minutes for that chosen $p$ .

Generalize back to expressions with $p$

Compute the ratio of each boundary value of $v$ to $p$ (for example, if $p=10$ and a boundary is $25$ , then the ratio is $2.5p$ ). Match these ratios to the form shown in the answer choices to identify which inequality correctly represents all possible $v$ values.

Step-by-step Explanation

Write an expression for the total monthly cost

The plan charges a fixed monthly texting fee of $p$ dollars plus $0.08$ dollars for each minute of voice calls.

So if the user talks for $v$ minutes, the total monthly cost $C$ (in dollars) is

C = p + 0.08v.

Translate the percent conditions into an inequality

"At least $120\%$ and at most $150\%$ of the monthly texting fee" means the total cost must be between $1.2p$ and $1.5p$ .

So we want

1.2p \le p + 0.08v \le 1.5p.

This is a compound inequality that we will solve for $v$ .

Isolate the term with $v$

Subtract $p$ from all three parts of the inequality:

1.2p - p \le p + 0.08v - p \le 1.5p - p.

Simplify each side:

0.2p \le 0.08v \le 0.5p.

Now the inequality is in terms of $0.08v$ . Next we solve for $v$ .

Solve for $v$ and match the choice

Divide all three parts of the inequality by $0.08$ (which is positive, so the inequality directions do not change):

\frac{0.2p}{0.08} \le v \le \frac{0.5p}{0.08}.

Compute the quotients:

$0.2 \div 0.08 = 2.5$
$0.5 \div 0.08 = 6.25$

So we get

2.5p \le v \le 6.25p,

which is exactly answer choice B.

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