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Solve the inequality |x + 4| − 3 < 5 and graph the solutions. Then write the solutions as a compound inequality. Thank you for helping me!!

User Arkerone
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1 Answer

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19 votes

Answer:

-12 < x < 4

see below for a graph

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.

Solve the inequality |x + 4| − 3 < 5 and graph the solutions. Then write the solutions-example-1
Solve the inequality |x + 4| − 3 < 5 and graph the solutions. Then write the solutions-example-2
User Crazyaboutliv
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2.9k points
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