# Question 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x+2-\frac{6}{x-3} \geq 0 \Rightarrow \frac{(x+3)(x-4)}{x-3} \geq 0$ | M1 | Attempt to combine fractions and factorise the numerator |
| $x=-3,\ x=4$ | A1, A1 | Correct critical values |
| $x \geq 4$ | A1ft | Follow through their 4 |
| $x=3$ | B1 | Identifies 3 as a critical value |
| $-3 \leq x < 3$ | M1A1 | M1: Attempt inside region; A1: Correct inequality |

**Way 2:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $6(x-3) \leq (x+2)(x-3)^2 \Rightarrow (x-3)(4-x)(x+3) \geq 0$ | M1 | Multiplies both sides by $(x-3)^2$ and attempt to factorise |
| $x=-3,\ x=4$ | A1, A1 | Correct critical values |
| $x \geq 4$ | A1ft | Follow through their 4 |
| $x=3$ | B1 | Identifies 3 as a critical value |
| $-3 \leq x < 3$ | M1A1 | M1: Attempt inside region; A1: Correct inequality |

**Way 3:**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x-3>0 \Rightarrow 6 \leq (x+2)(x-3) \Rightarrow (x-4)(x+3) \geq 0$ | M1 | Multiplies both sides by $(x-3)$, attempt to factorise, must state $x-3>0$ |
| $x=4$ | A1 | Correct critical value |
| $x \geq 4$ | A1ft | Follow through their 4 |
| $x=3$ | B1 | Identifies 3 as a critical value |
| $(x-3<0) \Rightarrow 6 \geq (x+2)(x-3),\ x=-3$ | A1 | Correct critical value |
| $(x+2)(x-3) \leq 6 \Rightarrow (x-4)(x+3) \leq 0 \Rightarrow -3 \leq x < 3$ | M1A1 | M1: Attempt inside region; A1: Correct inequality |

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