Suppose the inequality were #abs(4-x)+15 >14# instead of #abs(4 -x)+ 15 >21#. How would the solution change? Explain.?

1 Answer

Answer:

Because the absolute value function always returns a positive value, the solution changes from being some of the real numbers #(x< -2; x> 10)# to being all the real numbers #(x inRR)#

Explanation:

It looks like we're starting with the equation

#abs(4-x)+15>21#

We can subtract 15 from both sides and get:

#abs(4-x)+15color(red)(-15)>21color(red)(-15)#

#abs(4-x)>6#

at which point we can solve for #x# and see that we can have #x< -2; x> 10#

So now let's look at

#abs(4-x)+15>14#

and do the same with subtracting 15:

#abs(4-x)+15color(red)(-15)>14color(red)(-15)#

#abs(4-x)> -1#

Because the absolute value sign will always return a value that is positive, there is no value of #x# we can put into this inequality that will make #abs(4-x)<0#, let alone #-1#. And so the solution here is the set of all real numbers, which can be written #x inRR#