Question 6: Find the largest integer that satisfies each inequality below. \( \begin{array}{lll}\text { a) } x+3<9 & \text { (b) } 2 x+5<12 & \text { (c) } 7 x+10 \leq 31 \\ \text { d) } 3 x-5 \leq 9 & \text { (e) } \frac{x}{4}+3 \leq 8 & \text { (f) } 4 x+14 \leq 8\end{array} \)

To find the largest integer that satisfies each inequality, let's solve them one by one. a) \( x + 3 < 9 \) Subtract 3 from both sides: \( x < 6 \) The largest integer is **5**. b) \( 2x + 5 < 12 \) Subtract 5 from both sides: \( 2x < 7 \) Divide by 2: \( x < 3.5 \) The largest integer is **3**. c) \( 7x + 10 \leq 31 \) Subtract 10 from both sides: \( 7x \leq 21 \) Divide by 7: \( x \leq 3 \) The largest integer is **3**. d) \( 3x - 5 \leq 9 \) Add 5 to both sides: \( 3x \leq 14 \) Divide by 3: \( x \leq \frac{14}{3} \) or \( x \leq 4.67 \) The largest integer is **4**. e) \( \frac{x}{4} + 3 \leq 8 \) Subtract 3 from both sides: \( \frac{x}{4} \leq 5 \) Multiply by 4: \( x \leq 20 \) The largest integer is **20**. f) \( 4x + 14 \leq 8 \) Subtract 14 from both sides: \( 4x \leq -6 \) Divide by 4: \( x \leq -\frac{3}{2} \) or \( x \leq -1.5 \) The largest integer is **-1**. So, the largest integers for each inequality are: a) 5 b) 3 c) 3 d) 4 e) 20 f) -1