Solve the Absolute Value Equation: |4x+1| = 9

Q: Why does an absolute value equation have two solutions?

Because absolute value measures distance from zero! Both +9 and -9 are exactly 9 units away from zero, so $ |9| = |-9| = 9 $.

Q: Do I always get exactly two solutions?

Usually yes, but not always! Sometimes one solution doesn't work when you check it, and rarely both cases give the same answer. That's why checking is crucial.

Q: What if I get a decimal answer?

Decimal solutions are perfectly normal! In this problem, $ x = -2.5 $ is correct. Always double-check by substituting back into the original equation.

Q: Can I solve this by squaring both sides?

While squaring works mathematically, it's more complicated and can introduce extra solutions you need to eliminate. The two-case method is cleaner and more reliable for absolute value equations.

Absolute Value Equations with Integer Solutions

$\left|4x+1\right|=9$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left|4x+1\right|=9$

Step-by-step solution

To solve $\left|4x+1\right|=9$ , split into two scenarios:

1) $4x+1=9$ :

Subtract $1$ from both sides to get $4x=8$ .

Divide both sides by $4$ to find $x=2$ .

2) $4x+1=-9$ :

Subtract $1$ from both sides to get $4x=-10$ .

Divide both sides by $4$ to find $x=-2.5$ .

Thus, $x=2$ and $x=-2.5$ are solutions.

Final Answer

$x=2$ , $x=-2.5$

Key Points to Remember

Essential concepts to master this topic

Rule: Split absolute value equation into two separate cases
Technique: Set 4x+1 = 9 and 4x+1 = -9
Check: Both solutions must make original equation true: |4(2)+1| = |9| = 9 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't only solve 4x+1 = 9 and ignore the negative case = missing half the solutions! Absolute value equations almost always have two solutions because |a| = b means a = b OR a = -b. Always solve both 4x+1 = 9 AND 4x+1 = -9.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why does an absolute value equation have two solutions?

Because absolute value measures distance from zero! Both +9 and -9 are exactly 9 units away from zero, so $|9| = |-9| = 9$ .

Do I always get exactly two solutions?

Usually yes, but not always! Sometimes one solution doesn't work when you check it, and rarely both cases give the same answer. That's why checking is crucial.

How do I remember to solve both cases?

Think: "What two numbers have absolute value 9?" The answer is +9 and -9. So set the expression inside equal to both positive and negative values.

What if I get a decimal answer?

Decimal solutions are perfectly normal! In this problem, $x = -2.5$ is correct. Always double-check by substituting back into the original equation.

Can I solve this by squaring both sides?

While squaring works mathematically, it's more complicated and can introduce extra solutions you need to eliminate. The two-case method is cleaner and more reliable for absolute value equations.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Absolute Value and Inequality questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime