https://file.aiquickdraw.com/mathbot/file/17267737239777bgyixsb.jpg

We are asked to find the negative value of \( x \) that makes the equation true:

\[
10 + |3x + 6| = 30
\]

Let's solve this step by step.

### Step 1: Isolate the absolute value term
Subtract 10 from both sides of the equation:

\[
|3x + 6| = 30 - 10
\]

\[
|3x + 6| = 20
\]

### Step 2: Solve the absolute value equation
The absolute value equation \( |3x + 6| = 20 \) implies two possible cases:

1. \( 3x + 6 = 20 \)
2. \( 3x + 6 = -20 \)

#### Case 1: \( 3x + 6 = 20 \)
Solve for \( x \):

\[
3x = 20 - 6
\]
\[
3x = 14
\]
\[
x = \frac{14}{3} \quad \text{(not a negative value, so discard this solution)}
\]

#### Case 2: \( 3x + 6 = -20 \)
Solve for \( x \):

\[
3x = -20 - 6
\]
\[
3x = -26
\]
\[
x = \frac{-26}{3} = -\frac{26}{3}
\]

### Step 3: Match the solution to the choices
We convert \( -\frac{26}{3} \) into a mixed number:

\[
-\frac{26}{3} = -8 \frac{2}{3}
\]

This corresponds to option C.

### Final Answer:
The correct answer is **C.** \( -8 \frac{2}{3} \).

Would you like any further clarification? 

Here are 5 related questions to explore:
1. How do you solve other absolute value equations?
2. What is the method for solving inequalities involving absolute values?
3. Can you explain the importance of testing both cases in absolute value equations?
4. How do you convert improper fractions into mixed numbers?
5. How does the absolute value function affect graphing equations?

**Tip:** Always check both cases when solving absolute value equations, as they can lead to different solutions.

What negative value of x makes the equation 10 + |3x + 6| = 30 true?

Learn how to solve the absolute value equation 10 + |3x + 6| = 30. This detailed step-by-step solution walks through isolating the absolute value and solving for both positive and negative cases. Suitable for Grades 8-10 students studying algebra and absolute values.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation