2016 H2 Mathematics Paper 1 Question 1

Equations and Inequalities

Answers

3x2+5x−2x−4{\displaystyle \frac{ 3 x^2 + 5 x - 2 }{ x - 4 }}
x<−2   or   13<x<4{x < - 2 \; \textrm{ or } \; \frac{1}{3} < x < 4}

Full solutions

4x2+4x−14x−4−(x+3)=4x2+4x−14−(x+3)(x−4)x−4=4x2+4x−14−(x2−x−12)x−4=3x2+5x−2x−4  ■\begin{align*} & \frac{4 x^2 + 4 x - 14}{x - 4} - (x + 3) \\ & = \frac{4 x^2 + 4 x - 14-(x + 3)(x - 4)}{x - 4} \\ & = \frac{4 x^2 + 4 x - 14-(x^2 - x - 12)}{x - 4} \\ &= \frac{ 3 x^2 + 5 x - 2 }{ x - 4 } \; \blacksquare \end{align*}
4x2+4x−14x−4<x+34x2+4x−14x−4−(x+3)<03x2+5x−2x−4<0(x+2)(3x−1)x−4<0\begin{gather*} \frac{ 4 x^2 + 4 x - 14 }{ x - 4 } < x + 3 \\ \frac{ 4 x^2 + 4 x - 14 }{ x - 4 } - (x + 3) < 0 \\ \frac{ 3 x^2 + 5 x - 2 }{ x - 4 } < 0 \\ \frac{(x + 2)(3 x - 1)}{x - 4} < 0 \end{gather*}
x<−2   or   13<x<4  ■x < - 2 \; \textrm{ or } \; \frac{1}{3} < x < 4 \; \blacksquare