# Question 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{x-2}{2(x+2)} - \frac{12}{x(x+2)} \leq 0$ | M1 | Collects expressions to one side |
| $\frac{x^2-2x-24}{2x(x+2)} \leq 0$ | M1A1 | M1: Attempt common denominator. A1: Correct single fraction |
| $x = 0, -2$ | B1 | Correct critical values |
| $x^2-2x-24 \Rightarrow (x+4)(x-6)(=0) \Rightarrow x = \ldots$ | M1 | Attempt to solve quadratic as far as $x = \ldots$ |
| $x = -4, 6$ | A1 | Correct critical values. May be seen on a sketch |
| $-4 \leq x < -2,\ 0 < x \leq 6$ with $\leq$ or $<$ throughout | dM1A1 | M1: Attempt two inequalities using 4 CVs in ascending order (dependent on at least one previous M). A1: All 4 CVs in inequalities correct |
| $-4 \leq x < -2,\ 0 < x \leq 6$ or $[-4,-2) \cup (0,6]$ | A1cao **(9)** | Inequality signs correct. Set notation may use $\cup$ or "or" but not "and" |

**Alternative 1: Multiply both sides by $x^2(x+2)^2$**

| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2(x-2)(x+2) \leq 24x(x+2)$ | M1 | Both sides $\times x^2(x+2)^2$, must be a positive multiplier containing $x^2(x+2)^2$ |
| $x^3(x+2)-2x^2(x+2)-24x(x+2) \leq 0$ | M1A1 | M1: Collects to one side. A1: Correct inequality |
| $x = 0, -2$ | B1 | Correct critical values |
| $x^4-28x^2-48x = 0 \Rightarrow x(x+2)(x-6)(x+4)=0 \Rightarrow x = \ldots$ | M1 | Attempt to solve quartic; can cancel $x$ and $(x+2)$ and solve a quadratic |
| $x = -4, 6$ | A1 | Correct critical values |
| $-4 \leq x < -2,\ 0 < x \leq 6$ | dM1A1 | As above |
| $[-4,-2) \cup (0,6]$ | A1cao **(9)** | As above |

**Alternative 2: Using a sketch graph**

| Answer/Working | Mark | Guidance |
|---|---|---|
| Draw graphs of $y = \frac{x-2}{2(x+2)}$ and $y = \frac{12}{x(x+2)}$ | | |
| CVs $x = 0, -2$ | B1 | Vertical asymptotes of graphs |
| $\frac{x-2}{2(x+2)} = \frac{12}{x(x+2)}$ | M1 | Eliminate $y$ |
| $x(x-2) = 24$ | M1A1 | M1: Obtains quadratic. A1: Correct equation |
| $x^2-2x-24 \Rightarrow (x+4)(x-6)=0 \Rightarrow x = \ldots$ | M1 | Attempt to solve quadratic |
| CVs $x = -4, 6$ | A1 | Correct critical values |
| $-4 \leq x < -2,\ 0 < x \leq 6$ | dM1, A1, A1cao **(9)** | As main scheme |

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