Solve the Absolute Value Equation: |2x+3| = 9

Q: Why do absolute value equations have two solutions?

The absolute value $ |a| = 9 $ means the distance from zero is 9. Both +9 and -9 are exactly 9 units from zero, so both values work!

Q: How do I know when to split into two cases?

When you have $ |expression| = positive number $, always split! Set the expression equal to both the positive and negative of that number.

Q: Can absolute value equations have no solution?

Yes! If you have $ |expression| = negative number $, 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 depending on the equation.

Q: How can I check my answers quickly?

Substitute each answer back into the original equation. Both should make the absolute value equal to 9. If not, recheck your algebra!

Absolute Value Equations with Two Solutions

$\left|2x+3\right|=9$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|2x+3\right|=9$

Step-by-step solution

To solve the equation $\left|2x+3\right|=9$ , we split it into two separate equations:

1. $2x+3=9$

2. $2x+3=-9$

For the first equation:

$2x+3=9$

Subtract 3 from both sides:

$2x=6$

Divide both sides by 2:

$x=3$

For the second equation:

$2x+3=-9$

Subtract 3 from both sides:

$2x=-12$

Divide both sides by 2:

$x=-6$

Thus, the solutions are $x=3$ and $x=-6$ .

Final Answer

$x=3$ , $x=-6$

Key Points to Remember

Essential concepts to master this topic

Rule: Split absolute value equations into positive and negative cases
Technique: Solve 2x+3=9 and 2x+3=-9 separately for both solutions
Check: Substitute x=3: |2(3)+3|=|9|=9 ✓ and x=-6: |2(-6)+3|=|-9|=9 ✓

Common Mistakes

Avoid these frequent errors

Only solving the positive case
Don't just solve 2x+3=9 and stop = you miss half the solutions! Absolute value means the expression inside could be positive OR negative. Always set up both cases: 2x+3=9 AND 2x+3=-9.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why do absolute value equations have two solutions?

The absolute value $|a| = 9$ means the distance from zero is 9. Both +9 and -9 are exactly 9 units from zero, so both values work!

How do I know when to split into two cases?

When you have $|expression| = positive number$ , always split! Set the expression equal to both the positive and negative of that number.

What if I get the same answer twice?

That's possible! Sometimes both cases give the same solution. Just write it once in your final answer - don't repeat it.

Can absolute value equations have no solution?

Yes! If you have $|expression| = negative number$ , there's no solution because absolute values are never negative.

Do I always get exactly two solutions?

Not always! You might get two different solutions, one repeated solution, or no solution depending on the equation.

How can I check my answers quickly?

Substitute each answer back into the original equation. Both should make the absolute value equal to 9. If not, recheck your algebra!

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