Solve the Absolute Value Equation: |x| = 6

Q: Why does |x| = 6 have two answers?

Think of absolute value as distance from zero. Both 6 and -6 are exactly 6 units away from zero on the number line! So both values make the equation true.

Q: How do I know which solutions to include?

For equations like $ |x| = 6 $, always include both the positive and negative values. Check: $ |6| = 6 $ ✓ and $ |-6| = 6 $ ✓

Q: What if the right side is negative, like |x| = -3?

There would be no solution! Absolute values are always non-negative (zero or positive), so |x| can never equal a negative number.

Q: Do I need to write both answers in a special way?

Yes! Write your solution set as x = 6 or x = -6, or use set notation: {6, -6}. Both solutions are equally important.

Q: How can I visualize this on a number line?

Draw a number line and mark zero. Count 6 units to the right (positive 6) and 6 units to the left (negative 6). Both points are the same distance from zero!

Absolute Value Equations with Two Solutions

$\left|x\right|=6$

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Step-by-step written solution

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Understand the problem

$\left|x\right|=6$

Step-by-step solution

The equation is $\left|x\right|=6$ , so $x$ could be 6 or -6. Therefore, the solutions are $x=6$ and $x=-6$ .

Final Answer

Answers b + c

Key Points to Remember

Essential concepts to master this topic

Definition: Absolute value gives distance from zero, always positive
Method: For |x| = 6, consider both x = 6 and x = -6
Verification: Check both solutions: |6| = 6 ✓ and |-6| = 6 ✓

Common Mistakes

Avoid these frequent errors

Finding only the positive solution
Don't just write x = 6 and stop there = missing half the answer! Absolute value means distance from zero, and both positive and negative numbers can have the same distance. Always consider both x = a and x = -a when solving |x| = a.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why does |x| = 6 have two answers?

Think of absolute value as distance from zero. Both 6 and -6 are exactly 6 units away from zero on the number line! So both values make the equation true.

How do I know which solutions to include?

For equations like $|x| = 6$ , always include both the positive and negative values. Check: $|6| = 6$ ✓ and $|-6| = 6$ ✓

What if the right side is negative, like |x| = -3?

There would be no solution! Absolute values are always non-negative (zero or positive), so |x| can never equal a negative number.

Do I need to write both answers in a special way?

Yes! Write your solution set as x = 6 or x = -6, or use set notation: {6, -6}. Both solutions are equally important.

How can I visualize this on a number line?

Draw a number line and mark zero. Count 6 units to the right (positive 6) and 6 units to the left (negative 6). Both points are the same distance from zero!

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