Questions: Solve the absolute value equation. 3a+4=4a-2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is ∅.

$Solve the absolute value equation. 3a+4=4a-2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is ∅.$

Transcript text: Solve the absolute value equation. \[ |3 a+4|=|4 a-2| \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ \} . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The solution set is $\varnothing$.

Solution

Solution Steps

To solve the absolute value equation $|3a + 4| = |4a - 2|$, we need to consider the different cases where the expressions inside the absolute values are equal or opposite. This leads to two main cases to solve: $3a + 4 = 4a - 2$ and $3a + 4 = -(4a - 2)$. We then solve these linear equations separately and check if the solutions satisfy the original equation.

Step 1: Set Up the Equation

We start with the absolute value equation: \[ |3a + 4| = |4a - 2| \]

Step 2: Consider the Cases

To solve this, we consider two cases:

$3a + 4 = 4a - 2$
$3a + 4 = -(4a - 2)$

Step 3: Solve Case 1

For the first case: \[ 3a + 4 = 4a - 2 \] Subtract $3a$ from both sides: \[ 4 = a - 2 \] Add 2 to both sides: \[ a = 6 \]

Step 4: Solve Case 2

For the second case: \[ 3a + 4 = -(4a - 2) \] Distribute the negative sign: \[ 3a + 4 = -4a + 2 \] Add $4a$ to both sides: \[ 7a + 4 = 2 \] Subtract 4 from both sides: \[ 7a = -2 \] Divide by 7: \[ a = -\frac{2}{7} \]

Step 5: Combine the Solutions

The solutions from both cases are: \[ a = 6 \quad \text{and} \quad a = -\frac{2}{7} \]