Aladder is leaning against a building. the distance from the bottom of the ladder to the building is 18 ft less than the length of the ladder. how high up the side of the building is the top of the ladder if that distance is 1 ft less than the length of the ladder?

As you can see in the figure attached, to solve this exercise you must find the value of the length "L". Therefore, you need to follow the proccedure below:

 1. You must apply the Pythagorean Theorem, as below:

 a²=b²+c²

 a is the hypotenuse of the right triangle (a=L).
 b and c are the legs of the right triangle (b=L-18;  c=L-1)

 2. Then, you have:

 L²=(L-18)²+(L-1)²
 L²=[L²-2(L)(18)+(18)²]+[L²-2(L)(1)+(1)²]
 L²=L²-36L+324]+L²-2L+1²
 L²-38L+325=0

 3. As you can see, you obtain a quadratic equation. You can solve it by factoring. Then:

 (L-25)(L-13)=0

 L1=25
 L2=13

 4. You must choose L1=25, because if you decide to use L2=13, you will obtain a negative value in the base of the triangle and the distances can't be negative. Therefore:

 L=25 feet
 L-18=25-18=7 feet
 L-1=25-1=24 feet

 The answer is: 24 feet