Challenging +1.2 This is a Further Maths modulus inequality requiring systematic case analysis (checking when the expression inside is positive/negative), solving resulting quadratic inequalities, and careful consideration of domain restrictions (x ≠ -2) and the constraint 7-x > 0. While methodical, it's a standard F2 technique with multiple steps, placing it moderately above average difficulty.
Change sign of LHS or RHS and obtain equation (quadratic or linear, no simplification needed)
\(8x = -24 \Rightarrow\) CV \(-3\)
A1
Correct CV from solving the linear equation
\(x < -3 \quad -1 < x < 2\)
dddM1A1A1 (9)
dddM1: \(x <\) their smallest CV and \(x\) between their other 2 CVs. All M marks above needed. A1: Either inequality correct. A1: Both inequalities correct. "and" between inequalities acceptable. If \(\cap\) used, deduct an A mark
Total: [9]
# Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| [Sketch of graph] | — | This sketch on its own scores no marks, but may be seen in the work |
| $\frac{x^2+3x+10}{x+2} = 7-x \Rightarrow x^2+3x+10 = 14+5x-x^2$ | M1 | Form a quadratic equation or inequality, no simplification needed. **No algebra implies no marks** |
| $x^2-x-2=0 \Rightarrow (x-2)(x+1)=0$ | dM1 | Solve the 3TQ any valid method. Depends on first M mark |
| CVs $2, -1$ | A1A1 | A1: Either CV. A1: Both CVs |
| $\frac{-(x^2+3x+10)}{x+2} = 7-x \Rightarrow -x^2-3x-10=14+5x-x^2$ | M1 | Change sign of LHS or RHS and obtain equation (quadratic or linear, no simplification needed) |
| $8x = -24 \Rightarrow$ CV $-3$ | A1 | Correct CV from solving the linear equation |
| $x < -3 \quad -1 < x < 2$ | dddM1A1A1 (9) | dddM1: $x <$ their smallest CV and $x$ between their other 2 CVs. All M marks above needed. A1: Either inequality correct. A1: Both inequalities correct. "and" between inequalities acceptable. If $\cap$ used, deduct an A mark |
**Total: [9]**
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3. Use algebra to obtain the set of values of $x$ for which
$$\left| \frac { x ^ { 2 } + 3 x + 10 } { x + 2 } \right| < 7 - x$$
\hfill \mbox{\textit{Edexcel F2 2020 Q3 [9]}}