How do you graph and solve # abs(x + 4)<= -3#?

1 Answer
Lotusbluete
Dec 3, 2015

There are no solutions in #RR#.

Explanation:

So, let's start with graphing.

The graph of #|x|# looks like follows:

graph{|x| [-10.21, 9.79, -2.76, 7.24]}

It has the "peak" at #x = 0#.

To graph #|x+4|#, the peak needs to be at the solution of #x + 4 = 0 <=> x = -4#,
so the whole graph is basically shifted #4# points to the left:

graph{|x+4| [-10.21, 9.79, -2.76, 7.24]}

Now, the graphical representation of the inequality #|x+4| <= -3# are all the #x# values of those parts graph #|x+4|# with the #y# value smaller or equal than #-3#.

enter image source here

So, in my image those would be all the intervals where the blue graph is below the purple one.

As you can see, there are none, since the absolute value of any real number is always #>= 0# and can never be smaller or equal than # -3#.

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