Questions: Find the solution(s) to the following equation: 3x-9 = 9x + 5 Choose the correct answer below and fill in the values. Round to three decimal places if needed. x₁= or x₂= x₁= There is no solution.

Find the solution(s) to the following equation:
3x-9 = 9x + 5

Choose the correct answer below and fill in the values. Round to three decimal places if needed.
x₁= or x₂=
x₁=
There is no solution.

Transcript text: Find the solution(s) to the following equation: \[ |3 x-9|=9 x+5 \] Choose the correct answer below and fill in the values. Round to three decimal places if needed. $\mathrm{x}_{1}=$ $\square$ or $x_{2}=$ $\qquad$ $\mathrm{x}_{1}=$ $\square$ There is no solution.

Solution

Solution Steps

To solve the equation $|3x - 9| = 9x + 5$, we need to consider two cases due to the absolute value:

$3x - 9 = 9x + 5$
$3x - 9 = -(9x + 5)$

We will solve each case separately and then check if the solutions satisfy the original equation.

Step 1: Consider the Absolute Value Equation

To solve the equation $|3x - 9| = 9x + 5$, we need to consider two cases due to the absolute value:

$3x - 9 = 9x + 5$
$3x - 9 = -(9x + 5)$

Step 2: Solve the First Case

For the first case, we solve the equation: \[ 3x - 9 = 9x + 5 \] Rearranging terms, we get: \[ 3x - 9x = 5 + 9 \] \[ -6x = 14 \] \[ x = -\frac{14}{6} = -\frac{7}{3} \]

Step 3: Solve the Second Case

For the second case, we solve the equation: \[ 3x - 9 = -(9x + 5) \] Simplifying, we get: \[ 3x - 9 = -9x - 5 \] Rearranging terms, we get: \[ 3x + 9x = -5 + 9 \] \[ 12x = 4 \] \[ x = \frac{4}{12} = \frac{1}{3} \]

Step 4: Verify the Solutions

We need to verify if the solutions satisfy the original equation $|3x - 9| = 9x + 5$.

For $x = -\frac{7}{3}$: \[ |3(-\frac{7}{3}) - 9| = 9(-\frac{7}{3}) + 5 \] \[ |-7 - 9| = -21 + 5 \] \[ |-16| = -16 \quad \text{(False)} \]

For $x = \frac{1}{3}$: \[ |3(\frac{1}{3}) - 9| = 9(\frac{1}{3}) + 5 \] \[ |1 - 9| = 3 + 5 \] \[ |-8| = 8 \quad \text{(True)} \]