Solve the Absolute Value Equation: |x| = 4

Q: Why does |x| = 4 have two answers?

Absolute value measures distance from zero on the number line. Both 4 and -4 are exactly 4 units away from zero, so both are solutions!

Q: How do I know when to use positive or negative?

For equations like $ |x| = 4 $, you need both the positive and negative values. The absolute value 'removes' the sign, so both original values work.

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

That equation has no solution! Absolute values are always non-negative (zero or positive), so they can never equal a negative number.

Q: Do I always get exactly two solutions?

Not always! If $ |x| = 0 $, there's only one solution: x = 0. If the right side is negative, there are no solutions. Most other cases give two solutions.

Absolute Value Equations with Two Solutions

$\left|x\right|=4$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|x\right|=4$

Step-by-step solution

The equation is $\left|x\right|=4$ . This means that the value of $x$ can be either 4 or -4. Thus, the solutions are $x=4$ and $x=-4$ .

Final Answer

Answers a + c

Key Points to Remember

Essential concepts to master this topic

Definition: |x| = 4 means the distance from zero equals 4
Technique: Split into two cases: x = 4 and x = -4
Check: Verify both solutions: |4| = 4 and |-4| = 4 ✓

Common Mistakes

Avoid these frequent errors

Only finding the positive solution
Don't solve |x| = 4 as just x = 4! This gives only half the answer because absolute value strips away the sign. Always consider both positive and negative values that have the same distance from zero.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why does |x| = 4 have two answers?

Absolute value measures distance from zero on the number line. Both 4 and -4 are exactly 4 units away from zero, so both are solutions!

How do I know when to use positive or negative?

For equations like $|x| = 4$ , you need both the positive and negative values. The absolute value 'removes' the sign, so both original values work.

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

That equation has no solution! Absolute values are always non-negative (zero or positive), so they can never equal a negative number.

Do I always get exactly two solutions?

Not always! If $|x| = 0$ , there's only one solution: x = 0. If the right side is negative, there are no solutions. Most other cases give two solutions.

How can I check my work?

Substitute each answer back into the original equation. For our problem: $|4| = 4$ ✓ and $|-4| = 4$ ✓. Both should make the equation true!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Absolute Value and Inequality questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime