Solve the Absolute Value Equation: |x| = 3

Q: Why does |x| = 3 have two answers instead of one?

Because absolute value measures distance from zero! Both 3 and -3 are exactly 3 units away from zero on the number line, so both values make $ |x| = 3 $ true.

Q: How do I remember to find both solutions?

Think: "What two numbers have the same distance from zero?" Always write both equations: $ x = 3 $ and $ x = -3 $ when solving $ |x| = 3 $.

Q: What if the absolute value equals zero?

When $ |x| = 0 $, there's only one solution: x = 0. Zero is the only number that's exactly 0 units away from itself!

Q: Can absolute value equal a negative number?

Never! Absolute value is always positive or zero. If you see $ |x| = -5 $, the answer is no solution because distances can't be negative.

Absolute Value Equations with Dual Solutions

$\left|x\right|=3$

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Step-by-step written solution

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Understand the problem

$\left|x\right|=3$

Step-by-step solution

The equation is $\left|x\right|=3$ , which implies that $x$ can be 3 or -3. Hence, the solutions are $x=3$ and $x=-3$ .

Final Answer

Answers a + c

Key Points to Remember

Essential concepts to master this topic

Definition: Absolute value equals distance from zero on number line
Technique: Set up two equations: x = 3 and x = -3
Check: Verify both solutions: |3| = 3 and |-3| = 3 ✓

Common Mistakes

Avoid these frequent errors

Finding only one solution to absolute value equations
Don't solve |x| = 3 as just x = 3! This misses half the answer because absolute value represents distance, which is always positive. Always remember that |x| = a has two solutions: x = a and 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| = 3 have two answers instead of one?

Because absolute value measures distance from zero! Both 3 and -3 are exactly 3 units away from zero on the number line, so both values make $|x| = 3$ true.

How do I remember to find both solutions?

Think: "What two numbers have the same distance from zero?" Always write both equations: $x = 3$ and $x = -3$ when solving $|x| = 3$ .

What if the absolute value equals zero?

When $|x| = 0$ , there's only one solution: x = 0. Zero is the only number that's exactly 0 units away from itself!

Can absolute value equal a negative number?

Never! Absolute value is always positive or zero. If you see $|x| = -5$ , the answer is no solution because distances can't be negative.

How do I check my answers?

Substitute each solution back into the original equation. For $|x| = 3$ : check $|3| = 3$ ✓ and $|-3| = 3$ ✓

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