find t=for the interval

Question

The leg of a right triangle is 1 unit less than thrice its other leg. Its area should not be greater than 420 square units. What is the interval of possible values for the shorter leg? Let x be the shorter leg.

Answer 1

x< 12 just by trying a couple numbers, x= 10, y=30 is an area of 300

x=11, y= 33 is an area of 363, x=12, y= 36 gives an area of 432

that's a rough approximation

try x=12, y= 35, that gives exactly 420 = 12x35

so x<12. and x>0 thus

(0,12] is the interval of possible values for x the shorter leg of the right triangle, it can't be zero, but it could be 12. Or any number in between.

or use 2 inequalities to solve for x

x < 3y-1

area = xy/2 <420

you get a quadratic with 2 answers but one is negative and can be ignored.