Solve the Absolute Value Equation: |-3x| = 15

Q: Why do I need to consider two different cases?

Because absolute value measures distance from zero, which is always positive! For |-3x| = 15, the expression -3x could be either +15 or -15, since both have the same absolute value.

Q: How do I know which case to use first?

It doesn't matter which case you solve first! Set up both equations: -3x = 15 and -3x = -15. Solve each one separately to find all possible solutions.

Q: Can an absolute value equation have no solution?

Yes! If the right side is negative, there's no solution because absolute values are never negative. But since 15 is positive, this equation has solutions.

Q: What if I get the same answer for both cases?

Sometimes that happens! When it does, you have one repeated solution instead of two different ones. Always check both cases even if they look similar.

Q: Do I always get exactly two solutions?

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

Absolute Value Equations with Negative Coefficients

$\left|-3x\right|=15$

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Step-by-step written solution

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Understand the problem

$\left|-3x\right|=15$

Step-by-step solution

To solve $\left|-3x\right|=15$ , we consider both potential cases stemming from the absolute value:

1) $-3x=15$ :

Divide both sides by $-3$ to get $x=-5$ .

2) $-3x=-15$ :

Divide both sides by $-3$ to get $x=5$ .

Thus, the solutions are $x=-5$ and $x=5$ .

Final Answer

$x=-5$ , $x=5$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equations have two cases to consider
Technique: Set -3x = 15 and -3x = -15, then solve both
Check: Substitute both solutions: |-3(-5)| = 15 and |-3(5)| = 15 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the second case when solving absolute value equations
Don't solve only -3x = 15 to get x = -5 and stop there! This gives you only half the solution. The absolute value creates two scenarios. Always set up both cases: the expression equals the positive value AND the negative value.

Practice Quiz

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$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why do I need to consider two different cases?

Because absolute value measures distance from zero, which is always positive! For |-3x| = 15, the expression -3x could be either +15 or -15, since both have the same absolute value.

How do I know which case to use first?

It doesn't matter which case you solve first! Set up both equations: -3x = 15 and -3x = -15. Solve each one separately to find all possible solutions.

Can an absolute value equation have no solution?

Yes! If the right side is negative, there's no solution because absolute values are never negative. But since 15 is positive, this equation has solutions.

What if I get the same answer for both cases?

Sometimes that happens! When it does, you have one repeated solution instead of two different ones. Always check both cases even if they look similar.

Do I always get exactly two solutions?

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

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