Solve the Absolute Value Equation: -|x| = 15 Step-by-Step

Q: Why doesn't this equation have any solutions?

When we divide both sides by -1, we get $ |x| = -15 $. Since absolute values are always non-negative, there's no real number whose absolute value equals -15.

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

After isolating the absolute value, if it equals a negative number, there's no solution. Remember: $ |anything| ≥ 0 $ always!

Q: What if the equation was |x| = 15 instead?

Then you'd have two solutions: $ x = 15 $ and $ x = -15 $, because both numbers are 15 units away from zero.

Q: Should I always isolate the absolute value first?

Yes! Always get the absolute value by itself before solving. This helps you immediately see if a solution exists and avoid unnecessary work.

Q: How can I check that there's really no solution?

Try substituting any number into the original equation $ -|x| = 15 $. You'll find that $ -|x| $ is always zero or negative, never positive like 15!

Absolute Value Equations with Negative Coefficients

$-\left|x\right|=15$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$-\left|x\right|=15$

Step-by-step solution

Let's solve the equation $-\left|x\right|=15$ . Since the expression inside the absolute value can be either positive or negative, we consider two scenarios:

1. $-x = 15$ : This implies $x = -15$ .

2. $-(-x) = 15$ : This simplifies to $x = 15$ .

Thus, the solutions are $x = -15$ and $x = 15$ .

Final Answer

$x=-15$ , $x=15$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value expressions are always non-negative or zero
Technique: $-|x| = 15$ means $|x| = -15$ after dividing by -1
Check: Since $|x| \geq 0$ always, $|x| = -15$ is impossible ✓

Common Mistakes

Avoid these frequent errors

Incorrectly assuming absolute value equations always have two solutions
Don't automatically solve $-|x| = 15$ by setting up two cases like $x = ±15$ ! This ignores that absolute values cannot equal negative numbers. Always first isolate the absolute value expression and check if the result is non-negative.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why doesn't this equation have any solutions?

When we divide both sides by -1, we get $|x| = -15$ . Since absolute values are always non-negative, there's no real number whose absolute value equals -15.

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

After isolating the absolute value, if it equals a negative number, there's no solution. Remember: $|anything| ≥ 0$ always!

What if the equation was |x| = 15 instead?

Then you'd have two solutions: $x = 15$ and $x = -15$ , because both numbers are 15 units away from zero.

Should I always isolate the absolute value first?

Yes! Always get the absolute value by itself before solving. This helps you immediately see if a solution exists and avoid unnecessary work.

How can I check that there's really no solution?

Try substituting any number into the original equation $-|x| = 15$ . You'll find that $-|x|$ is always zero or negative, never positive like 15!

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