Solve for x: Understanding the Absolute Value in |x + 4| = 10

Q: Why do I need two equations for one absolute value equation?

Because absolute value measures distance from zero! Both +10 and -10 are exactly 10 units from zero, so x+4 could equal either value.

Q: How do I know which case to use first?

It doesn't matter! Solve both cases - the positive case (x+4=10) and the negative case (x+4=-10). You'll get two valid solutions.

Q: Can absolute value equations have no solutions?

Yes! If you get something like $ |x+3|=-5 $, there's no solution because absolute values are never negative.

Q: Do I always get exactly two solutions?

Not always! You could get two different solutions, one repeated solution, or no solution depending on the equation.

Absolute Value Equations with Integer Solutions

$\left|x+4\right|=10$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|x+4\right|=10$

Step-by-step solution

To solve the equation $\left|x+4\right|=10$ , we split it into two separate equations:

1. $x+4=10$

2. $x+4=-10$

For the first equation:

$x+4=10$

Subtract 4 from both sides:

$x=6$

For the second equation:

$x+4=-10$

Subtract 4 from both sides:

$x=-14$

Thus, the solutions are $x=6$ and $x=-14$ .

Final Answer

$x=6$ , $x=-14$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equations create two separate cases to solve
Technique: Split $|x+4|=10$ into x+4=10 and x+4=-10
Check: Substitute both solutions: $|6+4|=10$ and $|-14+4|=10$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't solve only x+4=10 and ignore x+4=-10 = missing half the solutions! The absolute value means the expression inside could equal positive OR negative 10. Always create both positive and negative cases.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why do I need two equations for one absolute value equation?

Because absolute value measures distance from zero! Both +10 and -10 are exactly 10 units from zero, so x+4 could equal either value.

How do I know which case to use first?

It doesn't matter! Solve both cases - the positive case (x+4=10) and the negative case (x+4=-10). You'll get two valid solutions.

Can absolute value equations have no solutions?

Yes! If you get something like $|x+3|=-5$ , there's no solution because absolute values are never negative.

What if I get the same answer from both cases?

That's possible! Sometimes both the positive and negative cases give the same solution, so you'll have just one answer instead of two.

Do I always get exactly two solutions?

Not always! You could get two different solutions, one repeated solution, or no solution depending on the equation.

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