DIRECTIONS: Use this information to answer Parts \( A \) and \( B \). Consider the absolute value equation \( a|x+b|+c=d \). Find an equation or inequality to describe when the equation has one solution and when the equa

To determine the conditions under which the absolute value equation \( a|x + b| + c = d \) has one solution or two solutions, we can analyze the equation step by step. ### Step 1: Isolate the Absolute Value Start by isolating the absolute value expression: \[ a|x + b| + c = d \quad \Rightarrow \quad a|x + b| = d - c \] Assuming \( a \neq 0 \), divide both sides by \( a \): \[ |x + b| = \frac{d - c}{a} \] ### Step 2: Analyze the Resulting Equation The equation \( |x + b| = k \) (where \( k = \frac{d - c}{a} \)) has different numbers of solutions based on the value of \( k \): 1. **Two Solutions**: If \( k > 0 \), the absolute value equation has two solutions. 2. **One Solution**: If \( k = 0 \), the equation has exactly one solution. 3. **No Solution**: If \( k < 0 \), there are no real solutions since the absolute value cannot be negative. ### Step 3: Translate Back to Original Variables Substituting back \( k = \frac{d - c}{a} \): - **Two Solutions**: \[ \frac{d - c}{a} > 0 \] This can occur in two scenarios: - \( d - c > 0 \) and \( a > 0 \) - \( d - c < 0 \) and \( a < 0 \) - **One Solution**: \[ \frac{d - c}{a} = 0 \quad \Rightarrow \quad d = c \] - **No Solution**: \[ \frac{d - c}{a} < 0 \] ### Summary - **One Solution**: The equation \( a|x + b| + c = d \) has exactly one solution **if and only if** \( d = c \). - **Two Solutions**: The equation has two distinct solutions **if and only if** \( \frac{d - c}{a} > 0 \). This condition holds when: - \( d > c \) and \( a > 0 \), **or** - \( d < c \) and \( a < 0 \). ### Example Let's consider an example to illustrate these conditions. **Given Equation**: \( 2|x - 3| + 4 = 10 \) 1. **Isolate the Absolute Value**: \[ 2|x - 3| = 10 - 4 \quad \Rightarrow \quad |x - 3| = 3 \] 2. **Determine Number of Solutions**: Since \( 3 > 0 \), there are two solutions: \[ x - 3 = 3 \quad \Rightarrow \quad x = 6 \] \[ x - 3 = -3 \quad \Rightarrow \quad x = 0 \] So, \( x = 0 \) and \( x = 6 \) are the two solutions. ### Conclusion By analyzing the equation \( a|x + b| + c = d \) and isolating the absolute value, we can determine the number of solutions based on the relationship between \( d \), \( c \), and \( a \). Specifically: - **One Solution** when \( d = c \). - **Two Solutions** when \( \frac{d - c}{a} > 0 \). This approach provides a clear method to assess the number of solutions for any absolute value equation of the given form.