Questions: 6+42x+6=14

6+42x+6=14
Transcript text: $6+4|2 x+6|=14$
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Solution

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Solution Steps

To solve the equation \(6 + 4|2x + 6| = 14\), we need to isolate the absolute value expression and then consider the two possible cases for the expression inside the absolute value (positive and negative). This will give us two separate linear equations to solve for \(x\).

Step 1: Isolate the Absolute Value Expression

The given equation is:

\[ 6 + 4|2x + 6| = 14 \]

First, subtract 6 from both sides to isolate the absolute value term:

\[ 4|2x + 6| = 14 - 6 \]

\[ 4|2x + 6| = 8 \]

Step 2: Solve for the Absolute Value

Divide both sides by 4 to solve for the absolute value:

\[ |2x + 6| = \frac{8}{4} \]

\[ |2x + 6| = 2 \]

Step 3: Solve the Absolute Value Equation

The equation \( |2x + 6| = 2 \) implies two separate equations:

  1. \( 2x + 6 = 2 \)
  2. \( 2x + 6 = -2 \)
Sub-step 3.1: Solve \( 2x + 6 = 2 \)

Subtract 6 from both sides:

\[ 2x = 2 - 6 \]

\[ 2x = -4 \]

Divide by 2:

\[ x = \frac{-4}{2} \]

\[ x = -2 \]

Sub-step 3.2: Solve \( 2x + 6 = -2 \)

Subtract 6 from both sides:

\[ 2x = -2 - 6 \]

\[ 2x = -8 \]

Divide by 2:

\[ x = \frac{-8}{2} \]

\[ x = -4 \]

Final Answer

The solutions to the equation are:

\[ \boxed{x = -2} \]

\[ \boxed{x = -4} \]

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