Mathenatices Solve \( 7(x+4)-3 / 3(x-6) \leqslant 2[x-3(x+5)] \)

To get started on solving the inequality \( 7(x+4)-3 / 3(x-6) \leqslant 2[x-3(x+5)] \), first simplify both sides. The left side can be rewritten: \[ \frac{7(x+4)-3}{3(x-6)} = \frac{7x + 28 - 3}{3(x-6)} = \frac{7x + 25}{3(x-6)} \] Then, simplify the right side: \[ 2[x - 3(x + 5)] = 2[x - 3x - 15] = 2[-2x - 15] = -4x - 30 \] Now, your inequality is: \[ \frac{7x + 25}{3(x-6)} \leqslant -4x - 30 \] To solve this, you'd multiply through by \( 3(x-6) \), remembering to flip the inequality if \( 3(x-6) \) is negative (which depends on the value of \( x \)). The resulting equation will allow you to find the critical values of \( x \). Analyze these values based on the signs of the expressions to sketch your solution on a number line. Happy solving! 🎉 When working through algebraic inequalities, one common mistake is forgetting to check if your multiplying factor is positive or negative, as this changes the direction of the inequality! Always break down the regions of your number line based on critical points to identify where the inequality holds true. Keep track of each step, as small errors can lead to different solutions. You’ve got this! 😊