How do you solve this system of equations: #x- 4y \geq 4; 4y - x > 8#?

The lines are parallel so there is no value #(x,y)# that will satisfy both equations.

They will never have a point in common because they will never intersect.

Solve both equations for #y# to put them in slope-intercept form

#x - 4y >=4#   Solve for #y#

1) Subtract #x# from both sides to isolate the #-4y# term
#- 4y >= -x + 4#

2) Divide both sides by #-4# to isolate #y#
#y >=(x)/(4) - 1#

#"Slope is"   (1)/(4)#

#color(white)(mmmmm)#―――――――

#4y - x > 8#   Solve for #y#

1) Add #x# to both sides to isolate the #4y# term
#4y > x + 8#

2) Divide all the terms on both sides by #4# to isolate #y#
#y > (x)/(4) - 2#

#"Slope is also"   (1)/(4)#

#color(white)(mmmmm)#―――――――

Because the slopes are the same, the lines are parallel.

That means that they will never have an intersection point, a point that satisfies both equations.

#color(white)(mmmmm)#―――――――

Check

Solve the equations as simultaneous equations to see if that gives a value for #y#

#x - 4y >=4#
#4y - x > 8#

- same as -

# - 4y + x >=  4#
#color(white)(m)##4y - x >   8#
#color(white)()#――――――――
#color(white)(.......)# #0    >= 12#

This is not true, so the equations are inconsistent.

#Check#