Help please I can't figure it out

Question

A family of lines have equations of the form y = −bx + c, where b and c are positive numbers.

For a line in this family that passes through the point(1,7):

find the values of c for which the x-axis intercept is less than or equal to 2.

(i know c ≤ (?) but i don't know how to work it out)

Answer 1

c must be greater than or = 14 for the x intercept to be < 2

y=-bx + c through the point (1,7)

7 = -b(1) + c

7=-b + c

b= c-7

y= -(c-7)x + c

x intercept < 2

x intercept = (x,0) x<2

0=-(c-7)x +c

0=-cx +7x + c

-cx +c = -7x

cx -c = 7x

c(x-1) =7x

c = 7x/(x-1)

if x = 2

c =7(2)/(2-1) = 14

c> 14 for x to be less than or equal to 2 when y=0

c is greater than or equal to 14 when the x intercept < 2

y=-bx+c, b=c-7, c=14

y = -(c-7)x +c

0 = -(14-7)2 +14

0 =-7(2) +14

0 = -14+14 = 0

if c<14 such as c =12

0= -(12-7)x+ 12

0 =--5x +12

x = 12/5 = 2.4 > 2

if c> 14 such as c =16

0 =-(16-7)x + 16

0= -9x +16

x = 16/9 = 1 7/9 < 2