Solve the Absolute Value Equation: |3x| = 21

Q: Why do I need to solve two equations for one absolute value?

Because absolute value makes both positive and negative numbers positive! The expression $ 3x $ could equal 21 or -21, and both would give $ |3x| = 21 $.

Q: How do I know which equation to solve first?

It doesn't matter! You can solve $ 3x = 21 $ or $ 3x = -21 $ first. Both equations are equally important and give you the complete solution set.

Q: Can absolute value equations have no solution?

Yes! If the absolute value equals a negative number, there's no solution because absolute values are always non-negative. For example, $ |3x| = -5 $ has no solution.

Q: How do I check if both answers are correct?

Substitute each answer back into the original equation. For $ x = 7 $: $ |3(7)| = |21| = 21 ✓ $. For $ x = -7 $: $ |3(-7)| = |-21| = 21 ✓ $

Absolute Value Equations with Integer Solutions

$\left|3x\right|=21$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|3x\right|=21$

Step-by-step solution

To solve the equation $\left|3x\right|=21$ , 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:

$3x = 21$ and $3x = -21$

3. Solving the first equation:

$3x = 21$

4. Divide both sides by 3:

$x = 7$

5. Solving the second equation:

$3x = -21$

6. Divide both sides by 3:

$x = -7$

7. Therefore, the solution is:

$x=-7$ , $x=7$

Final Answer

$x=-7$ , $x=7$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equations create two separate equations to solve
Technique: Set $3x = 21$ and $3x = -21$ then divide by 3
Check: Verify $|3(7)| = 21$ and $|3(-7)| = 21$ both work ✓

Common Mistakes

Avoid these frequent errors

Solving only one equation instead of both cases
Don't just solve $3x = 21$ and stop = missing half the solution! This ignores that the expression inside could be negative. Always set up both positive and negative cases for complete solutions.

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 solve two equations for one absolute value?

Because absolute value makes both positive and negative numbers positive! The expression $3x$ could equal 21 or -21, and both would give $|3x| = 21$ .

How do I know which equation to solve first?

It doesn't matter! You can solve $3x = 21$ or $3x = -21$ first. Both equations are equally important and give you the complete solution set.

What if I get the same answer from both equations?

That's rare but possible! If both cases give the same solution, you still have the correct answer. However, double-check your work because this usually means one equation was set up incorrectly.

Can absolute value equations have no solution?

Yes! If the absolute value equals a negative number, there's no solution because absolute values are always non-negative. For example, $|3x| = -5$ has no solution.

How do I check if both answers are correct?

Substitute each answer back into the original equation. For $x = 7$ : $|3(7)| = |21| = 21 ✓$ . For $x = -7$ : $|3(-7)| = |-21| = 21 ✓$

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