Solve the Absolute Value Equation: Finding X in |3x - 12| = 3

Q: Why does absolute value create two equations?

The absolute value |A| = B means the distance from zero equals B. This happens when A = B (positive case) or when A = -B (negative case). Both situations give the same distance!

Q: How do I remember which equation to write second?

Always write the negative version second: if you have |3x - 12| = 3, then write 3x - 12 = -3. Just put a negative sign in front of the number on the right side.

Q: Can absolute value equations have no solution?

Yes! If the right side is negative (like |x| = -5), there's no solution because absolute value is always non-negative. But when the right side is positive or zero, you'll get solutions.

Q: Do I always get exactly two solutions?

Not always! You get two solutions when the right side is positive, one solution when it equals zero, and no solution when it's negative.

Absolute Value Equations with Two Solutions

$\left|3x-12\right|=3$

❤️ Continue Your Math Journey!

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

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|3x-12\right|=3$

Step-by-step solution

To solve the equation $|3x - 12| = 3$ , follow these steps:

Step 1: Recognize that the absolute value equation $|A| = B$ means $A = B$ or $A = -B$ .
Step 2: Set up the two equations:

$3x - 12 = 3$
$3x - 12 = -3$

Now let's solve each equation:

For the first equation $3x - 12 = 3$ :
Add 12 to both sides to get:
$3x = 15$
Divide both sides by 3:
$x = 5$

For the second equation $3x - 12 = -3$ :
Add 12 to both sides to get:
$3x = 9$
Divide both sides by 3:
$x = 3$

Therefore, the solutions to the equation $|3x - 12| = 3$ are $x = 5$ and $x = 3$ .

These solutions correspond to choice 1: $x = 5$ , $x = 3$ .

Therefore, the solution to the problem is $x = 5$ and $x = 3$ .

Final Answer

$x=5$ , $x=3$

Key Points to Remember

Essential concepts to master this topic

Rule: |A| = B creates two equations: A = B and A = -B
Technique: 3x - 12 = 3 gives x = 5, and 3x - 12 = -3 gives x = 3
Check: Substitute both answers: |3(5) - 12| = 3 and |3(3) - 12| = 3 ✓

Common Mistakes

Avoid these frequent errors

Solving only one equation from the absolute value
Don't solve just 3x - 12 = 3 and stop = missing half the solutions! Absolute value creates two possible cases because the expression inside could be positive or negative. Always set up both A = B and A = -B equations.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why does absolute value create two equations?

The absolute value |A| = B means the distance from zero equals B. This happens when A = B (positive case) or when A = -B (negative case). Both situations give the same distance!

How do I remember which equation to write second?

Always write the negative version second: if you have |3x - 12| = 3, then write 3x - 12 = -3. Just put a negative sign in front of the number on the right side.

Can absolute value equations have no solution?

Yes! If the right side is negative (like |x| = -5), there's no solution because absolute value is always non-negative. But when the right side is positive or zero, you'll get solutions.

Do I always get exactly two solutions?

Not always! You get two solutions when the right side is positive, one solution when it equals zero, and no solution when it's negative.

What if my two solutions are the same number?

That happens when the right side equals zero! For example, |x - 5| = 0 gives you x = 5 twice. In this case, you have one unique solution.

🌟 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