How do you solve #-8(4x+6)< -24# and graph the solution on a number line?

Distribute the #-8# into #(4x+6)#.
#-32x-48<-24#

Add #-48# on both sides.
#-32x<24#

IMPORTANT: Since you're dividing both sides by a negative number (#-32#), the#<#will become a#># ---> #x>(-24/32)#

After simplifying the fraction, you will end up with #x>(-3/4)#.

How To Graph the Inequality
- Draw a number line such that it includes #-3/4#. You can count by one-fourths.
- Draw a point on #-3/4#. It should be an open (not filled) point because the inequality does not include #-3/4# as an answer; in other words, any value of x HAS to be greater than #-3/4#.
It would only be a closed point if the inequality is #x>=(-3/4)#.
- Draw a straight arrow pointing to the right that connects to the point at #-3/4#.

I hope this helps a lot! :)

Given: #-8(4x+6) < -24#

Two approaches the first step:

#color(brown)("Approach 1")#

Multiply both sides by (-1) to make everything positive and turn the inequality sign the other way round.

#+8(4x+6) > +24larr# the wide part of > points to #8(4x+6)#

#color(brown)("Approach 2")#

Given: #-8(4x+6) < -24#
As in the algebra shortcut method of changing sides of the = sign.

Move what is on the left of the #<# to its right and move what is on the right of < to its left. In doing so change their signs

#+24<+8(4x+6)larr# the wide part of > points to #8(4x+6)#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I choose the form: #+8(4x+6) > +24#

Divide both sides by 8

#4x+6>3#

Subtract 6 from both sides

#4x > -3#

Divide both sides by 4

#x> -3/4#

Number line 'graph'. The circle is hollow indicating 'greater than'.

Suppose it had been 'greater then or equal too'. In this case the circle would be filled in.

What is actually happening: The colored in area below #y=-24# is the feasible solution area for #(x,y)#

#x> -3/4#