Algebra (2) word problems?

Question

Ok. You need to use inequalities to solve these problems and I would like them worked out, please.

1. The usual toll charge to use the Bingham tunnel is 50 cents. If you purchase a special sticker for $5.50, the toll is only 35 cents. At least how many trips through the tunnel are needed before the sticker costs less than paying for each trip separately?

2. The lengths of the legs of an isosceles triangle are integers. The base is half as long as each leg. What are the possible lengths of the legs if the perimeter is between 6 units and 16 units?

Please work them out but I don't need too much extra info.

Thanks :)

Answer 1

.50x=5.50

x= 11 so one more 12 trips is $6.00

Answer 2

.50t>5.5 + .35t

.15t>5.5

t> 36.333, so if you do 37 trips or more, buy the pass

6 < 2l+1/2l < 16 where l = the length of the legs

6 < 5/2l < 16  multiply everything by 2/5

12/5 < L < 32/5

2 2/5 < L < 6 2/5

Answer 3

1) Let "t" be the number of trips you take.  Without buying the sticker, the total cost of the tolls after t trips would be 0.5 * t.  If you buy the sticker, then the total cost for everything is 5.50 plus 35 cents for each toll, or 5.5 + 0.35t.  We want to know how many tolls you need before the cost with the sticker is less than what it would have been without the sticker, so we want the t for which:

5.5 + 0.35 t < 0.5 t

Solve this for t, then round up to the nearest integer.

2) Since its an isosceles triangle, the two legs are the same.  Call this length L.  The base is half of this, or L/2.  So the perimeter is L + L + (L/2).  To make sure this sum is between 6 and 16, you need:

6 <= L + L + (L/2) <= 16

Simplify this to get just "L" in the middle.