A standard train ticket in a certain city costs $1.50 per ride. People who use the train also have the option of purchasing a frequent rider pass for $17.25 per month. With the pass, a ticket costs only $0.75 per ride. How many train rides a month make the frequent rider pass a better deal than standard train tickets?

X = number of rides per month.

Without the pass, the cost per ride is 2.00 per ride.

With the pass, the cost per ride is 1.25 per ride plus 15.75 per month.

You want to know at what number of rides does the cost per ride using the pass become cheaper than the cost per ride without using the pass.

The formula for total cost is as follows:

without the pass:

C1 = 2*x

with the pass:

C2 = 15.75 + 1.25*x

You want to know when C2 becomes less than C1.

C2 < C1 is the inequality equation you are looking for.

Since C2 = 15.75 + 1.25*x, and C1 = 2*x, this equation becomes:

15.75 + 1.25*x < 2*x

Subtract 1.25*x from both sides of this equation to get:

15.75 < 2*x - 1.25*x which becomes:

15.75 < .75*x

Divide both sides of this equation by .75*x to get:

15.75/.75 < x

Simplify to get:

21 < x

21 < x is the same as x > 21.

Your answer is the C2 becomes cheaper than C1 when x > 21.

If you make x = 21, then:

C1 = 2*21 = 42
C2 = 15.75 + 1.25*21 = 15.75 + 26.25 = 42

They are equal.

If you make x = 22, then:

C1 = 2*22 = 44
C2 = 15.75 + 1.25*22 = 15.75 + 27.5 = 43.25

43.25 is cheaper than 44 so C2 is cheaper than C1, confirming that the equation is good.