Solve the Absolute Value Equation: 2|x - 4| = 10

Q: Why do absolute value equations have two answers?

Because absolute value measures distance from zero! For example, both 5 and -5 are exactly 5 units from zero, so $|x| = 5$ has solutions x = 5 and x = -5.

Q: Do I always get exactly two solutions?

Not always! You might get two solutions, one solution, or no solutions. It depends on the equation. Always check both answers in the original equation to make sure they work.

Q: How do I know which operation to do first?

Always isolate the absolute value expression first before splitting into cases. In this problem, divide by 2 first to get $|x - 4| = 5$, then create your two cases.

Q: What if the right side is negative?

If you get something like $|x - 4| = -3$, there are no solutions! Absolute values are never negative, so the equation is impossible to solve.

Absolute Value Equations with Linear Expressions

$2|x - 4| = 10$

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Step-by-step written solution

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Understand the problem

$2|x - 4| = 10$

Step-by-step solution

To solve the equation $2|x - 4| = 10$ , divide both sides by 2. This gives $|x - 4| = 5$ .

The absolute value equation $|x - 4| = 5$ implies that $x - 4 = 5$ or $x - 4 = -5$ . Solving these equations:

1. $x - 4 = 5$ gives $x = 9$ .

2. $x - 4 = -5$ gives $x = -1$ .

Thus, the solutions are $x = 9$ and $x = -1$ .

Final Answer

Answers b + c

Key Points to Remember

Essential concepts to master this topic

Rule: Isolate absolute value first, then split into two cases
Technique: From $|x - 4| = 5$ , create x - 4 = 5 and x - 4 = -5
Check: Substitute both solutions: 2|9 - 4| = 10 and 2|-1 - 4| = 10 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case when solving absolute value
Don't solve only x - 4 = 5 and get just x = 9! This misses half the solutions because absolute value creates two cases. Always remember |A| = B means A = B OR A = -B, giving you two equations to solve.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do absolute value equations have two answers?

Because absolute value measures distance from zero! For example, both 5 and -5 are exactly 5 units from zero, so $|x| = 5$ has solutions x = 5 and x = -5.

Do I always get exactly two solutions?

Not always! You might get two solutions, one solution, or no solutions. It depends on the equation. Always check both answers in the original equation to make sure they work.

What if I get the same answer from both cases?

That's perfectly normal! Sometimes both the positive and negative cases give you the same solution. Just write it once as your final answer.

How do I know which operation to do first?

Always isolate the absolute value expression first before splitting into cases. In this problem, divide by 2 first to get $|x - 4| = 5$ , then create your two cases.

What if the right side is negative?

If you get something like $|x - 4| = -3$ , there are no solutions! Absolute values are never negative, so the equation is impossible to solve.

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