Solve the equation for all values of x.
open vertical bar, 3, x, plus, 8, close vertical bar, plus, 4, equals, x
∣3x+8∣+4=
x

We are given the equation:

\[
|3x + 8| + 4 = x
\]

### Step 1: Isolate the absolute value term

Subtract 4 from both sides:

\[
|3x + 8| = x - 4
\]

### Step 2: Split into two cases

The absolute value expression \( |3x + 8| \) results in two cases:

#### Case 1: \( 3x + 8 = x - 4 \)

1. Subtract \(x\) from both sides:
   \[
   3x - x + 8 = -4
   \]
   Simplifying:
   \[
   2x + 8 = -4
   \]
2. Subtract 8 from both sides:
   \[
   2x = -12
   \]
3. Divide by 2:
   \[
   x = -6
   \]

#### Case 2: \( 3x + 8 = -(x - 4) \)

1. Distribute the negative sign:
   \[
   3x + 8 = -x + 4
   \]
2. Add \(x\) to both sides:
   \[
   3x + x + 8 = 4
   \]
   Simplifying:
   \[
   4x + 8 = 4
   \]
3. Subtract 8 from both sides:
   \[
   4x = -4
   \]
4. Divide by 4:
   \[
   x = -1
   \]

### Step 3: Check for extraneous solutions

We need to check if the solutions satisfy the original equation.

1. For \(x = -6\):
   \[
   |3(-6) + 8| + 4 = -6 \implies |-18 + 8| + 4 = -6 \implies | -10| + 4 = -6 \implies 10 + 4 = 14 \neq -6
   \]
   So, \(x = -6\) is not a solution.

2. For \(x = -1\):
   \[
   |3(-1) + 8| + 4 = -1 \implies | -3 + 8| + 4 = -1 \implies |5| + 4 = -1 \implies 5 + 4 = 9 \neq -1
   \]
   So, \(x = -1\) is also not a solution.

### Final Conclusion

There are **no solutions** to the equation.

---

Would you like any further explanation, or do you have other questions? Here are 5 questions to deepen your understanding:

1. How can absolute value equations result in no solution?
2. What is the process for dealing with extraneous solutions?
3. Can you think of another example of an absolute value equation with no solution?
4. How would the equation change if the constant outside the absolute value was different?
5. What happens if the equation involves absolute value on both sides?

**Tip:** Always remember to check for extraneous solutions when solving absolute value equations, as they can sometimes lead to false results.

Solve the equation for all values of x. |3x + 8| + 4 = x

Learn how to solve the absolute value equation |3x + 8| + 4 = x with a detailed step-by-step explanation. This example covers absolute value properties and how to check for extraneous solutions. Suitable for students in Grades 8-10.

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