Solve the Absolute Value Equation: 2|x + 1| = 8

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

Because absolute value measures distance from zero! When $ |x + 1| = 4 $, the expression $ x + 1 $ could be 4 units away in either direction: +4 or -4.

Q: How do I know when to divide first before splitting into cases?

Always isolate the absolute value expression first! If you have $ 2|x + 1| = 8 $, divide by 2 to get $ |x + 1| = 4 $ before creating your two cases.

Q: Can an absolute value equation have no solutions?

Yes! If you get something like $ |x + 1| = -3 $, there's no solution because absolute values are never negative.

Q: Do I always get exactly two solutions?

Not always! You might get two different solutions, one repeated solution, or no solution at all. It depends on the specific equation.

Absolute Value Equations with Two-Step Isolation

Solve for $x$ in the equation $2|x + 1| = 8$ .

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Step-by-step written solution

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Understand the problem

Solve for $x$ in the equation $2|x + 1| = 8$ .

Step-by-step solution

Given the equation $2|x + 1| = 8$ , divide both sides by 2: $|x + 1| = 4$ .

Consider the two cases for the absolute value:

Case 1: $x + 1 = 4$
Subtract 1: $x = 3$ .
Case 2: $x + 1 = -4$
Subtract 1: $x = -5$ .

Thus, $x$ is $3 \text{ or } -5$ .

Final Answer

$x = 3 \text{ or } x = -5$

Key Points to Remember

Essential concepts to master this topic

Isolation Rule: Divide both sides to isolate the absolute value expression
Technique: Split into two cases: $x + 1 = 4$ and $x + 1 = -4$
Check: Substitute both solutions: $2|3 + 1| = 8$ and $2|-5 + 1| = 8$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case when solving absolute value equations
Don't just solve x + 1 = 4 and stop there = missing half the solutions! Absolute value means the expression inside could be positive OR negative. Always set up both cases: x + 1 = 4 AND x + 1 = -4.

Practice Quiz

Test your knowledge with interactive questions

Determine the absolute value of the following number:

$\left|18\right|=$

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! When $|x + 1| = 4$ , the expression $x + 1$ could be 4 units away in either direction: +4 or -4.

How do I know when to divide first before splitting into cases?

Always isolate the absolute value expression first! If you have $2|x + 1| = 8$ , divide by 2 to get $|x + 1| = 4$ before creating your two cases.

Can an absolute value equation have no solutions?

Yes! If you get something like $|x + 1| = -3$ , there's no solution because absolute values are never negative.

What if I get the same answer from both cases?

That's possible! Sometimes both cases lead to the same solution. Just write it once, not twice. Always check your work by substituting back into the original equation.

Do I always get exactly two solutions?

Not always! You might get two different solutions, one repeated solution, or no solution at all. It depends on the specific equation.

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