Answer:
-12 < x < 4
see below for a graph
Step-by-step explanation:
You want to solve the inequality |x + 4| − 3 < 5 and graph the solutions.
Solution
Adding 3 to both sides gives ...
|x +4| < 8
"Unfolding" this gives you ...
-8 < x +4 < 8
And subtracting 4 gives the compound inequality that is the solution:
-12 < x < 4
Graph
The graph of this compound inequality is shown in the first attachment. Open circles are used at the end points because they are not included in the solution set.
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Additional comment
The second attachment shows the inequality written in a form that can be easily compared to zero.
|x +4| -8 < 0 . . . . . . subtract 5 from both sides
You can readily identify the solution as being the x-values that correspond to that portion of the graph that is below the x-axis (f(x) < 0). Many graphing calculators identify the x-intercepts for you, so comparison to zero is made simple.
We know that the absolute value symbols mean ...
|x +4| < 8 ⇒ x +4 < 8 . . . when x +4 ≥ 0
|x +4| < 8 ⇒ -(x+4) < 8 . . . when x+4 < 0
The latter version can also be written ...
(x +4) > -8 when (x+4) < 0
This allows us to write the absolute value inequality as ...
-8 < x+4 < 8
where any solution to the left inequality requires x < -4, and any solution to the right inequality requires x ≥ -4. Those conditions give rise to the union of solution sets ...
x ∈ (-12, -4) ∪ [-4, 4)
You notice there is no gap between the parts of this solution, so we can write it as x ∈ (-12, 4) ⇔ -12 < x < 4.
In short, we sort of ignore the fact that we have to solve this on two different domains and merge the solutions.
