How do you solve #-6z-14>-32# and graph the solution on a number line?

First, solve for #z# almost the same way you would solve a 2-step equation:

Add #14# on both sides:

#-6z>46#

When you divide both sides by #-6#, you have to change the inequality sign:

#z<7.bar(6)#

This is what it looks like on a number line (notice the dot that is above #7.bar(6)# is open):

First, add #color(red)(14)# to each side of the inequality to isolate the #z# term while keeping the inequality balanced:

#-6z - 14 + color(red)(14) > 32 + color(red)(14)#

#-6z - 0 > 46#

#-6z > 46#

Now, divide each side of the inequality by #color(blue)(-6)# to solve for #z# while keeping the inequality balanced. However, because we are multiplying or dividing an inequality by a negative number we must reverse the inequality operator:

#(-6z)/color(blue)(-6) color(red)(<) 46/color(blue)(-6)#

#(color(blue)(cancel(color(black)(-6)))z)/cancel(color(blue)(-6)) color(red)(<) -23/3#

#z color(red)(<) -23/3#

Or

#z color(red)(<) -7 2/3#

To graph this we will draw an open circle at #-7 2/3# on the number line. The circle will be open because the inequality operator does not contain an "or equal to" clause.

We will the draw an arrow to the left of the circle because the inequality operator does contain a "less than" clause: