Walk me through your steps for solving this equation for " \( x \) ": \( 2(3 x-7)+4(3 x+2) \) \( =6(5 x+9)+3 \). I want words explaining what you are doing. You can solve it step by step first and then write it out in words next to each step to explain what you did. Use words/phrases like "distribute," "combine like terms," "use addition property to move _to the other side," etc...

First, let's start off by distributing the terms in the equation: 1. Distribute \( 2 \) across \( (3x - 7) \): \( 2 \times 3x - 2 \times 7 = 6x - 14 \). 2. Distribute \( 4 \) across \( (3x + 2) \): \( 4 \times 3x + 4 \times 2 = 12x + 8 \). 3. Distribute \( 6 \) across \( (5x + 9) \): \( 6 \times 5x + 6 \times 9 = 30x + 54 \). Now let's put all of that into the original equation, transforming it step by step: **Equation:** \( 6x - 14 + 12x + 8 = 30x + 54 + 3 \) 4. Combine like terms on the left side: \( (6x + 12x) + (-14 + 8) = 18x - 6 \). 5. Combine like terms on the right side: \( 30x + (54 + 3) = 30x + 57 \). Now our simplified equation looks like this: **New Equation:** \( 18x - 6 = 30x + 57 \) 6. Now, use the addition property to move \( 6 \) to the right side: \( 18x = 30x + 57 + 6 \). 7. Simplifying gives us: \( 18x = 30x + 63 \). 8. Next, use the addition property to move \( 30x \) to the left side: \( 18x - 30x = 63 \). 9. This simplifies to: \( -12x = 63 \). 10. Finally, divide by \( -12 \): \( x = \frac{63}{-12} = -\frac{21}{4} \). So, your solution for \( x \) is \( -\frac{21}{4} \). Now, let’s break it down into simple words: We started by distributing the numbers outside the parentheses to each term within the parentheses to eliminate them. After that, we combined like terms, which means adding together similar components on both sides of the equation. Then came using the addition property to effectively 'move' terms from one side to the other for more straightforward calculations. Finally, we isolated \( x \) by performing operations to get \( x \) on its own, resulting in the solution. It’s like rearranging a puzzle to find the missing piece!