An online service provider offers two monthly subscription plans. Under Plan A, the custom… — SAT

An online service provider offers two monthly subscription plans. Under Plan A, the customer pays a flat monthly fee of $C$ dollars. Under Plan B, the monthly cost, in dollars, is determined by the expression $1.5(20 - x) - 0.8(3x - 5)$ , where $x$ is the number of premium features the customer uses. The provider wants Plan B to be strictly cheaper than Plan A for any customer who uses more than 4 premium features. If $C$ is an integer, what is the minimum possible value of $C$ ?

Answer: 19 dollars

Answer

To find the minimum integer value of

C

, we simplify Plan B's cost expression to

34 - 3.9x

and set up the inequality

34 - 3.9x < C

. Solving for

x

by dividing by

-3.9

and reversing the inequality sign gives

x > \frac{34 - C}{3.9}

. For Plan B to be cheaper than Plan A for all customers using more than 4 features, the solution set

x > \frac{34 - C}{3.9}

must contain the interval

x > 4

. This requires the boundary point to be at most 4, so

\frac{34 - C}{3.9} \le 4

. Solving this inequality yields

C \ge 18.4

. The smallest integer value greater than or equal to

18.4

is 19.

Step-by-Step Solution

Simplify the cost expression for Plan B

34 - 3.9x

To combine like terms and express Plan B's cost in standard linear form.

Set up the inequality stating Plan B is strictly cheaper than Plan A

34 - 3.9x < C

Plan B is cheaper than Plan A when its cost is less than

C

dollars.

Solve the inequality for

x

in terms of

C

x > \frac{34 - C}{3.9}

Isolating

x

allows us to find the threshold number of premium features, remembering to reverse the inequality direction when dividing by the negative coefficient

-3.9

Relate the threshold condition to the given minimum number of premium features

\frac{34 - C}{3.9} \le 4

For Plan B to be cheaper for any

x > 4

, the solution interval

x > \frac{34 - C}{3.9}

must cover the entire interval

x > 4

. Thus, the boundary point must be at most 4.

Solve the boundary inequality for

C

C \ge 18.4

Multiplying by

3.9

and isolating

C

gives the lower bound for the cost of Plan A.

Find the minimum integer value for

C

Since

C

must be an integer and at least

18.4

, the smallest integer that satisfies this inequality is 19.

Key Concept

Solving linear inequalities in one variable with parameter constraints and real-world conditions.

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