Solve the Double Absolute Value Equation: -|-y²|

Q: Why doesn't the absolute value just cancel the negative sign?

The absolute value always makes expressions positive! Since $ -y^2 \leq 0 $, the absolute value changes it to $ y^2 $ (positive). Then the outer negative gives us $ -y^2 $.

Q: How do I know that -y² is always negative or zero?

Since $ y^2 \geq 0 $ for any real number y, then $ -y^2 \leq 0 $. This means negative y-squared is never positive, so the absolute value rule applies correctly.

Q: What if y = 0? Does the answer still work?

Yes! When y = 0, we have $ -y^2 = -0^2 = 0 $. The expression $ -|-y^2| = -|0| = -0 = 0 $, which still equals $ -y^2 $.

Q: Can I work from inside out with nested absolute values?

Absolutely! Always start with the innermost operation. Here: evaluate $ -y^2 $, then apply $ | \ | $, finally apply the outer negative sign.

Q: Why is this different from |-y|²?

The order matters! $ |-y|^2 = |y|^2 = y^2 $ (always positive), but $ -|-y^2| = -y^2 $ (always negative or zero). The position of the negative sign changes everything!

Absolute Value Properties with Negative Expressions

$-\left|-y^2\right|=$

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Understand the problem

$-\left|-y^2\right|=$

Step-by-step solution

To solve this problem, let's break it down into the following steps:

Step 1: Recognize that inside the absolute value, we have $-y^2$ . Since $y^2$ is non-negative, $-y^2$ is less than or equal to zero.
Step 2: Apply the absolute value: Since $-y^2 \leq 0$ , we use the property $|-a| = -(-a)$ , resulting in $\left|-y^2\right| = y^2$ .
Step 3: Apply the outer negative sign, $-\left|-y^2\right|$ , which simplifies to $-y^2$ .

Therefore, the solution to the problem is correctly identified as $-y^2$ .

Final Answer

$-y^2$

Key Points to Remember

Essential concepts to master this topic

Property: For any expression a ≤ 0, |a| equals -a
Technique: Since -y² ≤ 0, then |-y²| = -(-y²) = y²
Check: Test with y = 2: -|-4| = -4, which equals -y² ✓

Common Mistakes

Avoid these frequent errors

Assuming absolute value always removes negative signs
Don't think |-y²| = -y² directly! This skips the crucial step where absolute value makes negative expressions positive first. Always recognize that |-y²| = y² first, then apply the outer negative sign to get -y².

Practice Quiz

Test your knowledge with interactive questions

Determine the absolute value of the following number:

$\left|18\right|=$

FAQ

Everything you need to know about this question

Why doesn't the absolute value just cancel the negative sign?

The absolute value always makes expressions positive! Since $-y^2 \leq 0$ , the absolute value changes it to $y^2$ (positive). Then the outer negative gives us $-y^2$ .

How do I know that -y² is always negative or zero?

Since $y^2 \geq 0$ for any real number y, then $-y^2 \leq 0$ . This means negative y-squared is never positive, so the absolute value rule applies correctly.

What if y = 0? Does the answer still work?

Yes! When y = 0, we have $-y^2 = -0^2 = 0$ . The expression $-|-y^2| = -|0| = -0 = 0$ , which still equals $-y^2$ .

Can I work from inside out with nested absolute values?

Absolutely! Always start with the innermost operation. Here: evaluate $-y^2$ , then apply $| \ |$ , finally apply the outer negative sign.

Why is this different from |-y|²?

The order matters! $|-y|^2 = |y|^2 = y^2$ (always positive), but $-|-y^2| = -y^2$ (always negative or zero). The position of the negative sign changes everything!

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