Solving a System of Absolute Value Equations: When Does |x+2|=0?

Q: Why can't I just use x=-2 as my answer since it solves the first equation?

In a system of equations, the solution must satisfy every equation at the same time. While x=-2 works for $ |x+2|=0 $, it doesn't work for $ |x+2|=|x-2| $.

Q: What does 'no solution' actually mean?

It means there's no x-value that satisfies both equations simultaneously. Think of it like trying to be in two different places at the same time - it's impossible!

Q: Could I have made an error if I got x=0?

Yes! x=0 only satisfies $ |x+2|=|x-2| $ but not $ |x+2|=0 $. Always substitute your answer into every equation to verify.

Q: Is this different from solving one absolute value equation?

Absolutely! A system requires the same x-value to work in multiple equations. Single equations can have solutions that don't work for the entire system.

System of Equations with No Solution

$\begin{cases} |x+2|=|x-2| \\ \lvert x+2\rvert=0 \end{cases}$

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Understand the problem

$\begin{cases} |x+2|=|x-2| \\ \lvert x+2\rvert=0 \end{cases}$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Solve the equation $|x+2| = 0$
This implies $x+2 = 0$ , hence $x = -2$ .
Step 2: Solve the equation $|x+2| = |x-2|$ .
This leads to two cases:
- Case 1: $x+2 = x-2$ results in $2 = -2$ , which is false.
- Case 2: $x+2 = -(x-2)$ simplifies to $2x = 0$ , hence $x = 0$ .
Step 3: Check for intersection of solutions.
We have solutions $x = -2$ from the first equation and $x = 0$ from the absolute value equation.
Step 4: Verify if any common $x$ satisfies both equations.

Substituting $x = -2$ into $|x+2| = |x-2|$ , we find that it does not satisfy $|x-2| = |-4| = 4$ . Therefore, $x = -2$ is not a solution to the system.

Substituting $x = 0$ into both equations does not satisfy $|x+2| = 0$ . Therefore, there is no overlap of solutions.

The solution to the system is No solution.

Final Answer

No solution

Key Points to Remember

Essential concepts to master this topic

Absolute Value Rule: |x+2|=0 means x+2=0, so x=-2
System Method: Both equations must be satisfied simultaneously by same x-value
Verification: Check x=-2 in |x+2|=|x-2|: |0|≠|−4|, no solution ✓

Common Mistakes

Avoid these frequent errors

Solving each equation separately without checking intersection
Don't solve |x+2|=0 to get x=-2 and |x+2|=|x-2| to get x=0, then pick one = wrong answer! These are separate solutions to different equations. Always check if any x-value satisfies BOTH equations simultaneously.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't I just use x=-2 as my answer since it solves the first equation?

In a system of equations, the solution must satisfy every equation at the same time. While x=-2 works for $|x+2|=0$ , it doesn't work for $|x+2|=|x-2|$ .

What does 'no solution' actually mean?

It means there's no x-value that satisfies both equations simultaneously. Think of it like trying to be in two different places at the same time - it's impossible!

How do I know when a system has no solution?

Find all solutions to each equation separately, then check if any overlap. If no x-value satisfies all equations in the system, write 'No solution'.

Could I have made an error if I got x=0?

Yes! x=0 only satisfies $|x+2|=|x-2|$ but not $|x+2|=0$ . Always substitute your answer into every equation to verify.

Is this different from solving one absolute value equation?

Absolutely! A system requires the same x-value to work in multiple equations. Single equations can have solutions that don't work for the entire system.

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