Solve the System of Absolute Value Equations: |x-11|-1=5 and |x-5|=0

Q: Why should I solve |x-5|=0 first instead of the other equation?

The equation $ |x-5|=0 $ gives you exactly one solution immediately: x=5. The nested absolute value equation $ ||x-11|-1|=5 $ would create multiple cases to solve. Start simple!

Q: What does it mean when |x-5|=0?

When an absolute value equals zero, the expression inside must equal zero. So $ |x-5|=0 $ means $ x-5=0 $, giving us $ x=5 $.

Q: How do I handle the nested absolute value ||x-11|-1|?

Work from the inside out! First find $ |x-11| $, then subtract 1, then take the absolute value of that result. With x=5: $ |5-11|=6 $, then $ |6-1|=5 $.

Q: Could this system have multiple solutions?

No! Since $ |x-5|=0 $ has only one solution (x=5), the entire system can have at most one solution. The second equation just determines whether that solution works.

Absolute Value Equations with System Constraints

$\begin{cases} ||x-11|-1|=5 \\ |x-5|=0 \end{cases}$

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Understand the problem

$\begin{cases} ||x-11|-1|=5 \\ |x-5|=0 \end{cases}$

Step-by-step solution

To solve this system of absolute value equations, we follow these steps:

Step 1: Solve the equation $|x-5|=0$ .
Step 2: Validate the result with the other equation $||x-11|-1|=5$ .

Let's work through each step:

Step 1: Solve $|x-5|=0$ .
The equation $|x-5| = 0$ implies that the expression inside the absolute value must be zero. Therefore, $x-5=0$ gives us:

$x = 5$

Step 2: Validate with $||x-11|-1|=5$ .
Now, substitute $x = 5$ into the other equation:

$|5-11| = | -6 | = 6$

$|6-1| = |5| = 5$

This simplifies to $|5| = 5$ , which is true. Therefore, the solution $x = 5$ satisfies both equations.

Thus, the system of equations has a single solution: $x = 5$ .

Final Answer

$x=5$

Key Points to Remember

Essential concepts to master this topic

System Rule: Both equations must be satisfied by the same x-value
Strategy: Solve simpler equation first: |x-5|=0 gives x=5 directly
Verification: Check x=5 in nested absolute value: ||5-11|-1| = |6-1| = 5 ✓

Common Mistakes

Avoid these frequent errors

Solving the complex equation first instead of the simpler one
Don't start with ||x-11|-1|=5 and find multiple solutions = wasted time and confusion! The nested absolute value creates many cases to check. Always solve |x-5|=0 first since it gives exactly one value, then verify that value works.

Practice Quiz

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$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why should I solve |x-5|=0 first instead of the other equation?

The equation $|x-5|=0$ gives you exactly one solution immediately: x=5. The nested absolute value equation $||x-11|-1|=5$ would create multiple cases to solve. Start simple!

What does it mean when |x-5|=0?

When an absolute value equals zero, the expression inside must equal zero. So $|x-5|=0$ means $x-5=0$ , giving us $x=5$ .

How do I handle the nested absolute value ||x-11|-1|?

Work from the inside out! First find $|x-11|$ , then subtract 1, then take the absolute value of that result. With x=5: $|5-11|=6$ , then $|6-1|=5$ .

What if the equations had different solutions?

If substituting x=5 into the second equation didn't work, the system would have no solution. Systems require the same x-value to satisfy all equations simultaneously.

Could this system have multiple solutions?

No! Since $|x-5|=0$ has only one solution (x=5), the entire system can have at most one solution. The second equation just determines whether that solution works.

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