Solve the Equation: -|x| = 8 - Understanding Absolute Value

Q: Why can't absolute value equal a negative number?

By definition, absolute value measures distance from zero, which is always positive or zero. Since $ |x| \geq 0 $ for all real numbers, we have $ -|x| \leq 0 $.

Q: What does it mean when there's no solution?

It means the equation is impossible to satisfy with any real number. Just like asking "what number plus 1 equals 0.5?" - sometimes math equations have no answer!

Q: How do I recognize when an absolute value equation has no solution?

Look for these patterns: absolute value equals negative, or negative absolute value equals positive. These are mathematically impossible!

Q: What if the equation was |x| = 8 instead?

Then you'd have solutions! $ |x| = 8 $ means x = 8 or x = -8 because both numbers are 8 units from zero.

Q: Are there other equations like this with no solutions?

Yes! Any time you have $ -|expression| = positive number $, there's no solution. The negative makes it impossible.

Absolute Value Equations with Negative Results

$-\left|x\right|=8$

❤️ Continue Your Math Journey!

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

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$-\left|x\right|=8$

Step-by-step solution

Let's solve the equation $-\left|x\right|=8$ . The expression inside the absolute value can take two forms:

1. $-x = 8$ : This gives $x = -8$ .

2. $-(-x) = 8$ : This simplifies to $x = 8$ .

Therefore, the solutions are $x = -8$ and $x = 8$ .

Final Answer

$x=-8$ , $x=8$

Key Points to Remember

Essential concepts to master this topic

Property: Absolute value is always non-negative, never equals negative numbers
Analysis: Since $|x| \geq 0$ , then $-|x| \leq 0$ always
Check: Test if $-|x| = 8$ is possible: negative never equals positive ✓

Common Mistakes

Avoid these frequent errors

Incorrectly solving by removing the negative sign
Don't solve $-|x| = 8$ by turning it into $|x| = 8$ = x = ±8! This ignores the negative sign and gives impossible solutions. Always remember that $-|x|$ is always zero or negative, so it cannot equal a positive number.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't absolute value equal a negative number?

By definition, absolute value measures distance from zero, which is always positive or zero. Since $|x| \geq 0$ for all real numbers, we have $-|x| \leq 0$ .

What does it mean when there's no solution?

It means the equation is impossible to satisfy with any real number. Just like asking "what number plus 1 equals 0.5?" - sometimes math equations have no answer!

How do I recognize when an absolute value equation has no solution?

Look for these patterns: absolute value equals negative, or negative absolute value equals positive. These are mathematically impossible!

What if the equation was |x| = 8 instead?

Then you'd have solutions! $|x| = 8$ means x = 8 or x = -8 because both numbers are 8 units from zero.

Are there other equations like this with no solutions?

Yes! Any time you have $-|expression| = positive number$ , there's no solution. The negative makes it impossible.

🌟 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