$$  $$$$ \frac{\left\vert3a+12\right\vert}{9}=0 $$find the extraneous solutions

We are given the equation:

\[
\frac{|3a + 12|}{9} = 0
\]

### Step 1: Multiply both sides by 9 to eliminate the denominator:

\[
|3a + 12| = 0
\]

### Step 2: Solve for the expression inside the absolute value:

We know that the absolute value of a number is 0 if and only if the number itself is 0. Thus:

\[
3a + 12 = 0
\]

### Step 3: Solve for $a$:

\[
3a = -12
\]

\[
a = -4
\]

### Step 4: Check for extraneous solutions:

An extraneous solution typically arises when dealing with square roots, logarithms, or absolute values. In this case, since the equation involves an absolute value, we should verify if any solutions violate the original equation.

When $a = -4$, the original equation becomes:

\[
\frac{|3(-4) + 12|}{9} = \frac{| -12 + 12 |}{9} = \frac{0}{9} = 0
\]

This satisfies the original equation, meaning that $a = -4$ is not an extraneous solution.

### Conclusion:
There are no extraneous solutions for this equation. The valid solution is $a = -4$.

---

Would you like any further details or clarification? Here are some related questions you might want to explore:

1. What are extraneous solutions, and when do they typically occur?
2. How would the solution change if there were a different constant in place of 9?
3. How do you solve equations involving absolute values without extraneous solutions?
4. Can you graphically represent the behavior of absolute value equations?
5. How do extraneous solutions arise in square root equations?

**Tip**: When dealing with absolute value equations, always remember to check your solutions to avoid extraneous solutions.

$$ \frac{\left\vert 3a+12 \right\vert}{9}=0 $$ find the extraneous solutions

Learn how to solve the absolute value equation |3a + 12| / 9 = 0 step by step. Explore the concept of extraneous solutions and verify if any arise. This detailed guide is suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation