Questions: Solve the absolute value equation or indicate that the equation has no solution. 2x-3=7

Solve the absolute value equation or indicate that the equation has no solution. 2x-3=7
Transcript text: Solve the absolute value equation or indicate that the equation has no solution. \[ |2 x-3|=7 \]
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Solution

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Solution Steps

To solve the absolute value equation \(|2x - 3| = 7\), we need to consider the two possible cases for the expression inside the absolute value. The first case is when the expression is equal to 7, and the second case is when the expression is equal to -7. We will solve both linear equations separately to find the possible values of \(x\).

Step 1: Set Up the Equations

To solve the absolute value equation \(|2x - 3| = 7\), we consider two cases:

  1. \(2x - 3 = 7\)
  2. \(2x - 3 = -7\)
Step 2: Solve the First Equation

For the equation \(2x - 3 = 7\), add 3 to both sides to get: \[ 2x = 10 \] Then, divide both sides by 2: \[ x = 5 \]

Step 3: Solve the Second Equation

For the equation \(2x - 3 = -7\), add 3 to both sides to get: \[ 2x = -4 \] Then, divide both sides by 2: \[ x = -2 \]

Final Answer

\(\boxed{x = 5, -2}\)

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