Solve the Complex Equation: Modulus of z + 4 Equals 12

Q: Why are there two solutions to this equation?

Because absolute value always gives a positive result! Both 12 and -12 have an absolute value of 12, so z + 4 can equal either value.

Q: Can absolute value equations ever have no solutions?

Yes! If the right side is negative (like $ |z + 4| = -5 $), there are no solutions because absolute values are never negative.

Q: Is there a pattern to remember for these problems?

Absolute Value Equations with Two Solutions

$\left| z + 4 \right| = 12$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left| z + 4 \right| = 12$

Step-by-step solution

To solve $\left| z + 4 \right| = 12$ , consider the two cases:

1. $z + 4 = 12$ gives $z = 8$

2. $z + 4 = -12$ gives $z = -16$

Thus, the solutions are $z = 8$ and $z = -16$ .

Final Answer

$z = -16$ , $z = 8$

Key Points to Remember

Essential concepts to master this topic

Definition: |z + 4| = 12 means z + 4 equals 12 or -12
Method: Solve z + 4 = 12 gives z = 8, and z + 4 = -12 gives z = -16
Verification: Check both: |8 + 4| = 12 ✓ and |-16 + 4| = |-12| = 12 ✓

Common Mistakes

Avoid these frequent errors

Only finding one solution instead of two
Don't solve just z + 4 = 12 to get z = 8 only! This misses half the answer because absolute value creates two cases. Always set up both z + 4 = 12 AND z + 4 = -12 to find both solutions.

Practice Quiz

Test your knowledge with interactive questions

$\left|-x\right|=10$

FAQ

Everything you need to know about this question

Why are there two solutions to this equation?

Because absolute value always gives a positive result! Both 12 and -12 have an absolute value of 12, so z + 4 can equal either value.

How do I know which case to use first?

It doesn't matter which order you solve them! Just make sure you solve both cases: the positive case and the negative case.

What if one of my solutions doesn't work when I check?

Then you made an arithmetic error! For absolute value equations like this one, both solutions should always work when you substitute back into the original equation.

Can absolute value equations ever have no solutions?

Yes! If the right side is negative (like $|z + 4| = -5$ ), there are no solutions because absolute values are never negative.

Is there a pattern to remember for these problems?

If $|expression| = positive\ number$ , then two solutions
If $|expression| = 0$ , then one solution
If $|expression| = negative\ number$ , then no solutions

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