Moderate -0.8 This is a straightforward rational inequality requiring sign analysis of (3-x+4)/(x-4) > 0, which simplifies to (7-x)/(x-4) > 0. While it requires understanding of critical points and sign changes (x=4 and x=7), it's a standard single-step technique with no conceptual subtlety, making it easier than average but not trivial since students must remember x≠4.
Consideration of graph sketch or table of values/signs \(\Rightarrow 4 < x < 7\)
M1dep*, B2
One each for \(4 < x\) and \(x < 7\)
OR
Answer
Marks
Guidance
\(3 = x - 4 \Rightarrow x = 7\); \(x = 4\) (asymptote); Critical values at \(x = 7\) and \(x = 4\)
M1*
Identification of critical values at \(x = 7\) and \(x = 4\)
Consideration of graph sketch or table of values/signs; \(4 < x < 7\)
M1dep*, B2
One each for \(4 < x\) and \(x < 7\)
OR
Answer
Marks
Guidance
Consider inequalities arising from both \(x < 4\) and \(x > 4\)
M1*
Solving appropriate inequalities to their \(x > 7\) and \(x < 7\); \(4 < x < 7\)
M1dep*, B2
One for each \(4 < x\) and \(x < 7\), and no other solutions
[4]
# Question 4:
$\frac{3}{x-4} > 1 \Rightarrow 3(x-4) > (x-4)^2$ | M1* | Multiply through by $(x-4)^2$
$\Rightarrow 0 > x^2 - 11x + 28$ | |
$\Rightarrow 0 > (x-4)(x-7)$ | M1dep* | Factorise quadratic
$\Rightarrow 4 < x < 7$ | B2 | One each for $4 < x$ and $x < 7$
**OR**
$\frac{3}{x-4} - 1 > 0 \Rightarrow \frac{7-x}{x-4} > 0$ | M1* | Obtain single fraction $> 0$
Consideration of graph sketch or table of values/signs $\Rightarrow 4 < x < 7$ | M1dep*, B2 | One each for $4 < x$ and $x < 7$
**OR**
$3 = x - 4 \Rightarrow x = 7$; $x = 4$ (asymptote); Critical values at $x = 7$ and $x = 4$ | M1* | Identification of critical values at $x = 7$ and $x = 4$
Consideration of graph sketch or table of values/signs; $4 < x < 7$ | M1dep*, B2 | One each for $4 < x$ and $x < 7$
**OR**
Consider inequalities arising from both $x < 4$ and $x > 4$ | M1* |
Solving appropriate inequalities to their $x > 7$ and $x < 7$; $4 < x < 7$ | M1dep*, B2 | One for each $4 < x$ and $x < 7$, and no other solutions
**[4]**
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