Solving the equation $15=|135-60t|$ on $0\le{t}\le{5}$; two possible solutions

Question

Ann is driving along a highway that passes through Boston. Her distance $d$, in miles, from Boston is given by the equation $d=\vert{135-60t}$| where $t$ is the time in hours since the start of her trip and $0\le{t}\le{5}$. Determine when Ann will be exactly 15 miles from Boston.

I have solved the equation $15=|135-60t|$ and I found two solutions by the mean of the absolute value by solving the equation $15=135-60t$ and after the calculation is found that $t=2$; or by solving the equation $-15=135-60t$ and after solving the equation is found that $t=2.5$. I am not clear in understanding the meaning of this exercise. What does the sign minus in the second equation express? When is Ann exactly 15 miles from Boston? I have only found that $t=2$ hours or $t=2.5$ hours.

Would you help me please? Thank you in advance.

Answer 1

She's driving through Boston. She got there at time $t=2.25$ - two and a quarter hours. Fifteen minutes before that she was $15$ miles south on her way to town, fifteen minutes later she was north on her way to Maine.

To maintain that speed on her trip it must have happened in the middle of the night.

Answer 2

Sounds right. She will be fifteen miles away as she approaches Boston and again after she passes through and leaves Boston.

Answer 3

There are two times when she is 15 miles away from Boston. there are two values that have the same absolute value on the real number line x and -x, d=|135-60t| gets converted into:

$$15=135-60t$$ and $$-15=-135+60t$$ solving the first we get $$-120=-60t\implies 2=t$$ and the second one gives $$-150=60t\implies 2=t$$ EDIT of course by simply plugging in values we would get the table:

$$\begin {array}{c|c}d & t\\\hline 135 & 0 \\75 & 1\\15 &2 \\45 & 3 \end{array}$$

okay I messed up at first the point is t doesn't need to have only one solution.