Solving the Equation: x-10 When Absolute Value Equals Zero

Q: Why doesn't this equation have two solutions like other absolute value equations?

Great question! Most absolute value equations like $ |x-10|=5 $ have two solutions because both positive and negative values inside can create the same absolute value. But when the absolute value equals zero, only one value works: the expression inside must be exactly zero.

Q: How is this different from solving |x-10|=5?

When $ |x-10|=5 $, you get two cases: $ x-10=5 $ or $ x-10=-5 $. But when $ |x-10|=0 $, there's only one case: $ x-10=0 $ because zero is neither positive nor negative.

Q: What if I got confused and tried to solve x-10=±0?

That's a common mix-up! Remember that ±0 is just 0. Writing $ x-10=±0 $ gives you the same equation twice: $ x-10=0 $ and $ x-10=0 $, so you still get $ x=10 $ as the only solution.

Q: Can absolute value ever be negative?

Never! Absolute value represents distance, which is always zero or positive. If you see an equation like $ |x-10|=-3 $, it has no solution because absolute values can't be negative.

Absolute Value Equations with Zero Value

$\left|x-10\right|=0$

❤️ 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

$\left|x-10\right|=0$

Step-by-step solution

To solve the problem $\left|x-10\right|=0$ , we use this logical reasoning:

Step 1: Recognize that the absolute value expression equals zero.
The property of absolute values states that for $\left|A\right|=0$ , $A$ must equal zero.
Step 2: Apply this principle to the given equation.
Given $\left|x-10\right|=0$ , it follows that $x-10=0$ .
Step 3: Solve for $x$ .
Rearrange the equation $x - 10 = 0$ to find $x = 10$ .

The solution to the equation $\left|x-10\right|=0$ is therefore $x = 10$ .

Final Answer

$x=10$

Key Points to Remember

Essential concepts to master this topic

Rule: When absolute value equals zero, the expression inside equals zero
Technique: From $|x-10|=0$ , set $x-10=0$
Check: Substitute: $|10-10| = |0| = 0$ ✓

Common Mistakes

Avoid these frequent errors

Thinking absolute value equations always have two solutions
Don't assume |x-10|=0 has solutions x=10 and x=-10 = wrong reasoning! Most absolute value equations have two solutions, but when the absolute value equals zero, there's only one solution. Always remember that |A|=0 means A must equal exactly zero.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why doesn't this equation have two solutions like other absolute value equations?

Great question! Most absolute value equations like $|x-10|=5$ have two solutions because both positive and negative values inside can create the same absolute value. But when the absolute value equals zero, only one value works: the expression inside must be exactly zero.

How is this different from solving |x-10|=5?

When $|x-10|=5$ , you get two cases: $x-10=5$ or $x-10=-5$ . But when $|x-10|=0$ , there's only one case: $x-10=0$ because zero is neither positive nor negative.

What if I got confused and tried to solve x-10=±0?

That's a common mix-up! Remember that ±0 is just 0. Writing $x-10=±0$ gives you the same equation twice: $x-10=0$ and $x-10=0$ , so you still get $x=10$ as the only solution.

Can absolute value ever be negative?

Never! Absolute value represents distance, which is always zero or positive. If you see an equation like $|x-10|=-3$ , it has no solution because absolute values can't be negative.

How do I check my answer is correct?

Substitute your answer back into the original equation. For $x=10$ : $|10-10| = |0| = 0$ ✓. The left side equals the right side, so your answer is correct!

🌟 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