Questions: Solve for all values of (b) in simplest form. [ 9=15+b ]

Transcript text: Solve for all values of $b$ in simplest form. \[ 9=|15+b| \]

Solution

Solution Steps

To solve the equation $9 = |15 + b|$, we need to consider the definition of absolute value. The absolute value equation $ |x| = a $ implies two possible equations: $ x = a $ or $ x = -a $. Therefore, we will set up two separate equations: $ 15 + b = 9 $ and $ 15 + b = -9 $. Solving these equations will give us the possible values for $ b $.

Step 1: Set Up the Absolute Value Equation

We start with the equation given in the problem: \[ 9 = |15 + b| \] This implies two cases based on the definition of absolute value.

Step 2: Solve the First Case

For the first case, we set up the equation: \[ 15 + b = 9 \] To isolate $ b $, we subtract 15 from both sides: \[ b = 9 - 15 \] This simplifies to: \[ b = -6 \]

Step 3: Solve the Second Case

For the second case, we set up the equation: \[ 15 + b = -9 \] Again, we isolate $ b $ by subtracting 15 from both sides: \[ b = -9 - 15 \] This simplifies to: \[ b = -24 \]

Final Answer

The solutions for $ b $ are: \[ \boxed{b = -6} \quad \text{and} \quad \boxed{b = -24} \]

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