Standard +0.8 This requires systematic case analysis of the modulus inequality, solving the quadratic x²+x-2=0 to find critical points, then solving two separate quadratic inequalities and combining solutions. More demanding than standard modulus questions due to the quadratic expression inside the modulus and the need to carefully track multiple solution intervals.
Obtain and solve a 3TQ (any valid method including calculator)
CVs \(x = \frac{-1 \pm \sqrt{73}}{4}\)
A1
2 correct CVs. Allow decimal equivalents (1.886..., −2.386...), min 3 sf
\(-x^2 - x + 2 < \frac{1}{2}x + \frac{5}{2}\)
M1
Multiply either side by \(-1\)
\(2x^2 + 3x + 1 > 0\), \((2x+1)(x+1) > 0\)
M1
Obtain and solve a 3TQ
CVs \(x = -\frac{1}{2},\ -1\)
A1
2 correct CVs
\(\frac{-1-\sqrt{73}}{4} < x < -1,\quad -\frac{1}{2} < x < \frac{-1+\sqrt{73}}{4}\)
M1, A1
Form 2 double inequalities with CVs, no overlap; correct inequality signs. Values must be exact (note 0.5 is exact). Allow "and" but not "\(\cap\)". May use "\(\cup\)" with round brackets
## Question 3:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $2x^2 + x - 9 < 0$ | M1 | Obtain and solve a 3TQ (any valid method including calculator) |
| CVs $x = \frac{-1 \pm \sqrt{73}}{4}$ | A1 | 2 correct CVs. Allow decimal equivalents (1.886..., −2.386...), min 3 sf |
| $-x^2 - x + 2 < \frac{1}{2}x + \frac{5}{2}$ | M1 | Multiply either side by $-1$ |
| $2x^2 + 3x + 1 > 0$, $(2x+1)(x+1) > 0$ | M1 | Obtain and solve a 3TQ |
| CVs $x = -\frac{1}{2},\ -1$ | A1 | 2 correct CVs |
| $\frac{-1-\sqrt{73}}{4} < x < -1,\quad -\frac{1}{2} < x < \frac{-1+\sqrt{73}}{4}$ | M1, A1 | Form 2 double inequalities with CVs, no overlap; correct inequality signs. Values must be exact (note 0.5 is exact). Allow "and" but not "$\cap$". May use "$\cup$" with round brackets |
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3. Use algebra to obtain the set of values of $x$ for which
$$\left| x ^ { 2 } + x - 2 \right| < \frac { 1 } { 2 } ( x + 5 )$$
\hfill \mbox{\textit{Edexcel F2 2021 Q3 [7]}}