First, add #color(red)(54)# to each segment of the system of inequalities to isolate the #h# term while keeping the system balanced:
#-22 + color(red)(54) < -23h - 54 + color(red)(54) < -15 + color(red)(54)#
#32 < -23h - 0 < 39#
#32 < -23h < 39#
Next, divide each segment by #color(blue)(-23)# to solve for #h# while keeping the system balanced. However, because we are multiplying or dividing inequalities by a negative number we must reverse the inequality operators:
#32/color(blue)(-23) color(red)(>) (-23h)/color(blue)(-23) color(red)(>) 39/color(blue)(-23)#
#32/-23 color(red)(>) (color(blue)(cancel(color(black)(-23)))h)/cancel(color(blue)(-23)) color(red)(>) 39/-23#
#-32/23 color(red)(>) h color(red)(>) -39/23#
Or
#h > -39/23# and #h < -32/23#
Or, in interval notation:
#(-39/23, -32/23)#
