lional Equations Question 5, 1.2 Paticl2 The following rational equation has denominators that contain variables For llis equation Keeping the restrictions in mind, solve the equation. \[ \frac{3}{x}=\frac{9}{8 x}+3 \] a. What is/are the value or values of the variable that make(s) the denominators zero? \[ x= \] \( \square \) (Simplify your answer Uso a comma to separate answers as needed)

To find the values of the variable that make the denominators zero, we need to set the denominators in the equation to zero. For your equation, the denominators are \(x\) and \(8x\). Setting these equal to zero gives us: 1. \(x = 0\) 2. \(8x = 0\) (which simplifies to \(x = 0\)) So, the value that makes the denominators zero is: \[ x=0 \] \( \square \) Now, let's dive into why knowing this is important! If you want to avoid reaching mathematical dead ends, it's crucial to pay attention to those pesky values that can make a denominator zero, as they can cause your equation to be undefined. Working with rational equations without addressing these restrictions could lead you to slap your forehead in frustration when the solution doesn’t check out! Also, solving rational equations is like a puzzle where you want to carefully maneuver through the steps without dropping pieces. Make sure to always isolate variables first and transform your equation before you dive into the calculations. Don't skip steps or you might mistakenly add, subtract, or divide by a variable that could vanish into zero!