Solve the following inequalities, illustrating each solution on a number line. \begin{tabular}{|l|l|}\hline (a) \( 2 x>18 \) and \( 3 x-2<40 \) & (b) \( x+3 \leq 5 \) and \( 2(1-5 x)<32 \)\end{tabular}

Let's solve each inequality step by step! **(a) Solve \( 2x > 18 \) and \( 3x - 2 < 40 \)** 1. For \( 2x > 18 \): - Divide both sides by 2: \[ x > 9 \] So, the solution is \( x > 9 \). 2. For \( 3x - 2 < 40 \): - Add 2 to both sides: \[ 3x < 42 \] - Divide by 3: \[ x < 14 \] So, the solution is \( x < 14 \). Combining these two results, the solution for part (a) is: \[ 9 < x < 14 \] **Number Line:** - Draw a line with open circles on 9 and 14 (showing that both are not included) and shade the region between them. --- **(b) Solve \( x + 3 \leq 5 \) and \( 2(1 - 5x) < 32 \)** 1. For \( x + 3 \leq 5 \): - Subtract 3 from both sides: \[ x \leq 2 \] So, the solution is \( x \leq 2 \). 2. For \( 2(1 - 5x) < 32 \): - Divide both sides by 2: \[ 1 - 5x < 16 \] - Subtract 1 from both sides: \[ -5x < 15 \] - Divide by -5 (remember to flip the inequality): \[ x > -3 \] So, the solution is \( x > -3 \). Combining these results, the solution for part (b) is: \[ -3 < x \leq 2 \] **Number Line:** - Draw a line with an open circle at -3 (not included) and a closed circle at 2 (included), shading the region between -3 and 2.