Questions: 12 = 3 + 3x

Transcript text: $12=|3+3 x|$

Solution

Solution Steps

To solve the equation $12 = |3 + 3x|$, we need to consider the two possible cases for the absolute value expression. The first case is when the expression inside the absolute value is positive or zero, and the second case is when it is negative. We will solve for $x$ in both scenarios and check which solutions satisfy the original equation.

Step 1: Understand the Absolute Value Equation

The given equation is:

\[ 12 = |3 + 3x| \]

An absolute value equation of the form $ |A| = B $ implies two possible equations:

$ A = B $
$ A = -B $

Step 2: Set Up the Two Possible Equations

For the given equation $ |3 + 3x| = 12 $, we set up the two possible equations:

$ 3 + 3x = 12 $
$ 3 + 3x = -12 $

Step 3: Solve the First Equation

Solve the equation $ 3 + 3x = 12 $:

\[ 3 + 3x = 12 \]

Subtract 3 from both sides:

\[ 3x = 12 - 3 \]

\[ 3x = 9 \]

Divide both sides by 3:

\[ x = \frac{9}{3} \]

\[ x = 3 \]

Step 4: Solve the Second Equation

Solve the equation $ 3 + 3x = -12 $:

\[ 3 + 3x = -12 \]

Subtract 3 from both sides:

\[ 3x = -12 - 3 \]

\[ 3x = -15 \]

Divide both sides by 3:

\[ x = \frac{-15}{3} \]

\[ x = -5 \]

Final Answer

The solutions to the equation $ 12 = |3 + 3x| $ are:

\[ \boxed{x = 3} \]

\[ \boxed{x = -5} \]

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