Question

Solve the absolute value equation 6−2|3x+2|−9=−11
6
−
2
|
3
𝑥
+
2
|
−
9
=
−
11
.(1 point)
Responses

x=23
𝑥
=
2
3
x equals Start Fraction 2 over 3 End Fraction

x=−5
𝑥
=
−
5
and x=113
𝑥
=
11
3
x equals negative 5 and x equals Start Fraction 11 over 3 End Fraction

x=−2
𝑥
=
−
2
and x=23
𝑥
=
2
3
x equals negative 2 and x equals Start Fraction 2 over 3 End Fraction

no solution

Answer 1

To solve the equation \( 6 - 2|3x + 2| - 9 = -11 \), we first simplify the left-hand side:

1. Combine constants on the left:
\[
6 - 9 = -3
\]
This simplifies our equation to:
\[
-3 - 2|3x + 2| = -11
\]

2. Now, isolate the absolute value term:
\[
-2|3x + 2| = -11 + 3
\]
\[
-2|3x + 2| = -8
\]

3. Divide both sides by -2:
\[
|3x + 2| = 4
\]

Now we have two cases to solve for \( |3x + 2| = 4 \):

### Case 1:
\[
3x + 2 = 4
\]
Subtract 2 from both sides:
\[
3x = 2
\]
Now, divide by 3:
\[
x = \frac{2}{3}
\]

### Case 2:
\[
3x + 2 = -4
\]
Subtract 2 from both sides:
\[
3x = -6
\]
Now, divide by 3:
\[
x = -2
\]

### Final Solutions:
The solutions to the equation \( 6 - 2|3x + 2| - 9 = -11 \) are:
\[
x = \frac{2}{3} \quad \text{and} \quad x = -2
\]

Thus, the correct response is:
\[
x = -2 \quad \text{and} \quad x = \frac{2}{3}
\]