Question 1208939
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Answers: <font color=red>x = -1 and x = 5</font>


Explanation


The rule we need is |x| = k breaks down into x = k or x = -k where k is nonnegative.
For instance, |x| = 27 means x = 27 or x = -27. 
Both -27 and 27 are the same distance from zero on the number line.


Using that rule we can break
|3x - |2x+1|| = 4
into
3x - |2x+1| = 4 and 3x - |2x+1| = -4


Let's solve the first piece.
3x - |2x+1| = 4
|2x+1| = 3x-4
2x+1 = 3x-4 or 2x+1 = -(3x-4)
2x+1 = 3x-4 or 2x+1 = -3x+4
2x-3x = -4-1 or 2x+3x = 4-1
-x = -5 or 5x = 3
x = 5 or x = 3/5
Those are two potential solutions so far. 
The key term is "potential" since we need to verify each solution.


Now solve the 2nd piece found earlier.
3x - |2x+1| = -4
|2x+1| = 3x+4
2x+1 = 3x+4 or 2x+1 = -(3x+4)
2x+1 = 3x+4 or 2x+1 = -3x-4
2x-3x = 4-1 or 2x+3x = -4-1
-x = 3 or 5x = -5
x = -3 or x = -5/5 = -1
We found two more potential solutions.


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The four potential solutions we need to check are
x = 5 or x = 3/5
x = -3 or x = -1


Let's return to the original equation and plug in x = 5
|3x - |2x+1|| = 4
|3*5 - |2*5+1|| = 4
|15 - |10+1|| = 4
|15 - |11|| = 4
|15 - 11| = 4
|4| = 4
4 = 4 ..... works
The solution <font color=red>x = 5</font> is confirmed.


Now test x = 3/5
|3x - |2x+1|| = 4
|3*3/5 - |2*3/5+1|| = 4
|9/5 - |6/5+1|| = 4
|9/5 - |6/5+5/5|| = 4
|9/5 - |11/5|| = 4
|9/5 - 11/5| = 4
|(9-11)/5| = 4
|-2/5| = 4
2/5 = 4 ..... this doesn't work out
We determine that x = 3/5 doesn't work, so it is extraneous.
We must cross it off the list.


Now onto x = -3
|3x - |2x+1|| = 4
|3*(-3) - |2*(-3)+1|| = 4
|-9 - |-6+1|| = 4
|-9 - |-5|| = 4
|-9 - 5| = 4
|-14| = 4
14 = 4 ...... this doesn't work
Remove x = -3 from the list.


Lastly we need to check x = -1
|3x - |2x+1|| = 4
|3*(-1) - |2*(-1)+1|| = 4
|-3 - |-2+1|| = 4
|-3 - |-1|| = 4
|-3 - 1| = 4
|-4| = 4
4 = 4 ...... this works


Therefore we confirm that only <font color=red>x = -1 and x = 5</font> are the answers.


Verification using a Desmos graph is shown here
<a href="https://www.desmos.com/calculator/lr9kd0nk5f">https://www.desmos.com/calculator/lr9kd0nk5f</a>
The two intersection points have x coordinates of x = -1 and x = 5. 
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