Solve the Absolute Value Equation: |-4x| = 12

Q: Why do I need to create two separate equations?

Because absolute value measures distance from zero, which is always positive! The expression inside $ |-4x| $ could equal 12 or -12, and both give the same absolute value of 12.

Q: Can absolute value ever equal a negative number?

Never! Absolute value is always zero or positive. If you see something like $ |x| = -5 $, there's no solution because absolute values can't be negative.

Q: Do I always get exactly two solutions?

Not always! You could get two different solutions (like this problem), one solution (if both equations give the same answer), or no solution (if the absolute value equals a negative number).

Absolute Value Equations with Linear Expressions

$\left|-4x\right|=12$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|-4x\right|=12$

Step-by-step solution

To solve the equation $\left|-4x\right|=12$ , we consider the definition of absolute value:

1. The expression inside the absolute value can be either positive or negative, but its absolute value is always positive.

2. Therefore, we set up two equations to solve:

$-4x = 12$ and $-4x = -12$

3. Solving the first equation:

$-4x = 12$

4. Divide both sides by -4:

$x = -3$

5. Solving the second equation:

$-4x = -12$

6. Divide both sides by -4:

$x = 3$

7. Therefore, the solution is:

$x=-3$ , $x=3$

Final Answer

$x=-3$ , $x=3$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equals positive number creates two opposite equations
Technique: Set -4x = 12 and -4x = -12, then solve each
Check: Substitute both solutions: |-4(-3)| = 12 and |-4(3)| = 12 ✓

Common Mistakes

Avoid these frequent errors

Only solving one equation instead of two
Don't solve just -4x = 12 and stop there = missing half the solutions! Absolute value creates two possibilities because the expression inside can be positive or negative. Always set up both equations: one positive and one negative.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why do I need to create two separate equations?

Because absolute value measures distance from zero, which is always positive! The expression inside $|-4x|$ could equal 12 or -12, and both give the same absolute value of 12.

How do I know which equation to write first?

It doesn't matter which order you write them! Just make sure you have both cases: $-4x = 12$ and $-4x = -12$ . The order won't change your final answers.

What if I get the same answer from both equations?

That's possible! Sometimes absolute value equations have only one unique solution. But you still need to check both cases to be sure you haven't missed anything.

Can absolute value ever equal a negative number?

Never! Absolute value is always zero or positive. If you see something like $|x| = -5$ , there's no solution because absolute values can't be negative.

Do I always get exactly two solutions?

Not always! You could get two different solutions (like this problem), one solution (if both equations give the same answer), or no solution (if the absolute value equals a negative number).

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