Solve the System: Equalizing |x+2| and |x-2| with x⁴ = 0

Q: Why should I solve the polynomial equation first?

The equation $ x^4 = 0 $ has only one solution ($ x = 0 $), while $ |x+2| = |x-2| $ could have multiple solutions. Starting with the most restrictive equation saves time!

Q: How do I solve absolute value equations like |x+2| = |x-2|?

Absolute value equations $ |A| = |B| $ mean either A = B or A = -B. So $ x+2 = x-2 $ (impossible) or $ x+2 = -(x-2) $ which gives $ x = 0 $.

Q: What if the polynomial equation had no real solutions?

If $ x^4 = 0 $ had no solutions, then the entire system would have no solutions! In systems of equations, all equations must be satisfied simultaneously.

Q: Can I check my answer by plugging into both equations?

Absolutely! For $ x = 0 $: Check $ 0^4 = 0 ✓ $ and $ |0+2| = |0-2| $ gives $ 2 = 2 ✓ $. Both work!

Q: Why are the other answer choices wrong?

Test them! For $ x = 1 $: $ 1^4 = 1 ≠ 0 $ ✗. For $ x = 2 $: $ 2^4 = 16 ≠ 0 $ ✗. Only $ x = 0 $ satisfies both equations.

Systems of Equations with Polynomial Constraints

$\begin{cases} |x+2|=|x-2| \\ x^4=0 \end{cases}$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\begin{cases} |x+2|=|x-2| \\ x^4=0 \end{cases}$

Step-by-step solution

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

Step 1: Solve the polynomial equation $x^4 = 0$ .
Step 2: Verify the solution against the absolute value equation $|x+2| = |x-2|$ .
Step 3: Ensure the solution matches a provided choice, if necessary.

Now, let's work through each step:

Step 1: Solving $x^4 = 0$ yields $x = 0$ , as any non-zero values to the power of 4 would not equal zero.

Step 2: Substituting $x = 0$ into the absolute value equation:

$|0+2| = |0-2|$

$|2| = |-2|$

$2 = 2$ , thus the condition is satisfied.

Step 3: Compare with provided answer choices. Since the correct choice must satisfy both equations, $x = 0$ is consistent with option (1).

Therefore, the solution to the problem is $x = 0$ .

Final Answer

$x=0$

Key Points to Remember

Essential concepts to master this topic

Polynomial Rule: Solve $x^4 = 0$ first to get $x = 0$
Verification Method: Substitute $x = 0$ into $|0+2| = |0-2|$ gives $2 = 2$
System Check: Solution must satisfy both equations simultaneously for validity ✓

Common Mistakes

Avoid these frequent errors

Solving the absolute value equation first and ignoring the polynomial constraint
Don't solve $|x+2| = |x-2|$ to get $x = 0$ and stop there = incomplete solution! The polynomial equation $x^4 = 0$ is actually more restrictive and should be solved first. Always identify which equation provides the strongest constraint and solve that one first.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why should I solve the polynomial equation first?

The equation $x^4 = 0$ has only one solution ( $x = 0$ ), while $|x+2| = |x-2|$ could have multiple solutions. Starting with the most restrictive equation saves time!

How do I solve absolute value equations like |x+2| = |x-2|?

Absolute value equations $|A| = |B|$ mean either A = B or A = -B. So $x+2 = x-2$ (impossible) or $x+2 = -(x-2)$ which gives $x = 0$ .

What if the polynomial equation had no real solutions?

If $x^4 = 0$ had no solutions, then the entire system would have no solutions! In systems of equations, all equations must be satisfied simultaneously.

Can I check my answer by plugging into both equations?

Absolutely! For $x = 0$ : Check $0^4 = 0 ✓$ and $|0+2| = |0-2|$ gives $2 = 2 ✓$ . Both work!

Why are the other answer choices wrong?

Test them! For $x = 1$ : $1^4 = 1 ≠ 0$ ✗. For $x = 2$ : $2^4 = 16 ≠ 0$ ✗. Only $x = 0$ satisfies both equations.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Absolute Value and Inequality questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime