Solve the Absolute Value Equation: |2x + 1| = 9

Q: Why do I need to solve two separate equations?

Because absolute value measures distance from zero, which is always positive! So $ |2x + 1| = 9 $ means the expression inside could be either +9 or -9 units from zero.

Q: How do I know which case is positive and which is negative?

For $ |expression| = number $, always set up: expression = +number and expression = -number. The absolute value equation automatically gives you both cases!

Q: Can absolute value equations have no solutions?

Yes! If the right side is negative (like $ |x| = -3 $), there are no solutions because absolute value is never negative.

Q: Do I always get exactly two solutions?

Not always! You might get two different solutions, one repeated solution, or no solutions depending on the equation. Always solve both cases to find out!

Absolute Value Equations with Linear Expressions

$\left| 2x + 1 \right| = 9$

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Step-by-step written solution

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Understand the problem

$\left| 2x + 1 \right| = 9$

Step-by-step solution

To solve $\left| 2x + 1 \right| = 9$ , we consider two cases:

1. $2x + 1 = 9$ implies $2x = 8 \to x = 4$

2. $2x + 1 = -9$ implies $2x = -10 \to x = -5$

Thus, the solutions are $x = 4$ and $x = -5$ .

Final Answer

$x = 4$ , $x = -5$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equals positive means two cases to solve
Technique: Set 2x + 1 = 9 and 2x + 1 = -9
Check: Substitute both answers: |2(4) + 1| = 9 and |2(-5) + 1| = 9 ✓

Common Mistakes

Avoid these frequent errors

Solving only the positive case
Don't just solve 2x + 1 = 9 and stop = missing half the solutions! This ignores that absolute value creates two equal distances from zero. Always solve both the positive and negative cases.

Practice Quiz

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$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why do I need to solve two separate equations?

Because absolute value measures distance from zero, which is always positive! So $|2x + 1| = 9$ means the expression inside could be either +9 or -9 units from zero.

How do I know which case is positive and which is negative?

For $|expression| = number$ , always set up: expression = +number and expression = -number. The absolute value equation automatically gives you both cases!

What if I get the same answer for both cases?

That's possible! Sometimes both cases lead to the same solution. Just make sure to check your work by substituting back into the original equation.

Can absolute value equations have no solutions?

Yes! If the right side is negative (like $|x| = -3$ ), there are no solutions because absolute value is never negative.

Do I always get exactly two solutions?

Not always! You might get two different solutions, one repeated solution, or no solutions depending on the equation. Always solve both cases to find out!

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