Solve |x| = 7: Finding Values in an Absolute Value Equation

Q: Why does |x| = 7 have two answers?

Because absolute value measures distance from zero! Both 7 and -7 are exactly 7 units away from zero on the number line, so both make $ |x| = 7 $ true.

Q: How do I know when an absolute value equation has two solutions?

When the absolute value equals a positive number, there are always two solutions. If it equals zero, there's one solution. If it equals a negative number, there are no solutions.

Q: What if I only find x = 7 as my answer?

You're missing half the solution! Remember that two different numbers can have the same absolute value. Always ask yourself: "What negative number also has this absolute value?"

Q: How do I check my answers?

Substitute each solution back into the original equation: $ |7| = 7 $ ✓ and $ |-7| = 7 $ ✓. Both should give you the same result as the right side.

Q: Can absolute value equations have no solutions?

Yes! If you get something like $ |x| = -5 $, there are no solutions because absolute values are never negative. Distance can't be negative!

Absolute Value Equations with Two Solutions

$\left|x\right|=7$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|x\right|=7$

Step-by-step solution

The equation is $\left|x\right|=7$ , which means that the absolute value of $x$ is 7. Therefore, $x$ could be 7 or -7. Thus, the solutions are $x=7$ and $x=-7$ .

Final Answer

$x=-7, x=7$

Key Points to Remember

Essential concepts to master this topic

Definition: Absolute value represents distance from zero on number line
Technique: $|x| = 7$ means x = 7 or x = -7
Check: Verify $|7| = 7$ and $|-7| = 7$ both equal 7 ✓

Common Mistakes

Avoid these frequent errors

Finding only one solution instead of both
Don't solve $|x| = 7$ as just x = 7! This misses half the solutions because absolute value equations typically have two answers. Always consider both the positive and negative values that give the same absolute value.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why does |x| = 7 have two answers?

Because absolute value measures distance from zero! Both 7 and -7 are exactly 7 units away from zero on the number line, so both make $|x| = 7$ true.

How do I know when an absolute value equation has two solutions?

When the absolute value equals a positive number, there are always two solutions. If it equals zero, there's one solution. If it equals a negative number, there are no solutions.

What if I only find x = 7 as my answer?

You're missing half the solution! Remember that two different numbers can have the same absolute value. Always ask yourself: "What negative number also has this absolute value?"

How do I check my answers?

Substitute each solution back into the original equation: $|7| = 7$ ✓ and $|-7| = 7$ ✓. Both should give you the same result as the right side.

Can absolute value equations have no solutions?

Yes! If you get something like $|x| = -5$ , there are no solutions because absolute values are never negative. Distance can't be negative!

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