Questions: 4x-1 = 2x + 13

4x-1 = 2x + 13
Transcript text: 13. $|4 x-1|=2 x+13$
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Solution

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Solution Steps

To solve the equation \( |4x - 1| = 2x + 13 \), we need to consider two cases due to the absolute value:

  1. \( 4x - 1 = 2x + 13 \)
  2. \( 4x - 1 = -(2x + 13) \)

For each case, solve for \( x \) and check if the solutions satisfy the original equation.

Step 1: Consider the Absolute Value Cases

To solve the equation \( |4x - 1| = 2x + 13 \), we need to consider two cases due to the absolute value:

  1. \( 4x - 1 = 2x + 13 \)
  2. \( 4x - 1 = -(2x + 13) \)
Step 2: Solve the First Case

For the first case, solve the equation: \[ 4x - 1 = 2x + 13 \]

Subtract \( 2x \) from both sides: \[ 2x - 1 = 13 \]

Add 1 to both sides: \[ 2x = 14 \]

Divide by 2: \[ x = 7 \]

Step 3: Solve the Second Case

For the second case, solve the equation: \[ 4x - 1 = -(2x + 13) \]

Distribute the negative sign: \[ 4x - 1 = -2x - 13 \]

Add \( 2x \) to both sides: \[ 6x - 1 = -13 \]

Add 1 to both sides: \[ 6x = -12 \]

Divide by 6: \[ x = -2 \]

Step 4: Verify Solutions

Verify each solution by substituting back into the original equation:

  • For \( x = 7 \): \[ |4(7) - 1| = 2(7) + 13 \] \[ |28 - 1| = 14 + 13 \] \[ 27 = 27 \] (True)

  • For \( x = -2 \): \[ |4(-2) - 1| = 2(-2) + 13 \] \[ |-8 - 1| = -4 + 13 \] \[ |-9| = 9 \] \[ 9 = 9 \] (True)

Both solutions satisfy the original equation.

Final Answer

\(\boxed{x = 7, -2}\)

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