Standard +0.8 This requires systematic case analysis of modulus inequalities with a parameter, testing critical points x = a/2 and x = -3a, then solving resulting linear inequalities in each region. The presence of the parameter a adds algebraic complexity beyond standard modulus inequality questions, requiring careful manipulation to express the solution set in terms of a.
EITHER: State or imply non-modular inequality \(2^2(2x-a)^2 < (x+3a)^2\), or corresponding quadratic equation, or pair of linear equations \(2(2x-a) = \pm(x+3a)\)
B1
Make reasonable attempt at solving a 3-term quadratic, or solve two linear equations for \(x\)
M1
Obtain critical values \(x = \dfrac{5}{3}a\) and \(x = -\dfrac{1}{5}a\)
A1
State final answer \(-\dfrac{1}{5}a < x < \dfrac{5}{3}a\)
A1
OR: Obtain critical value \(x = \dfrac{5}{3}a\) from a graphical method, or by inspection, or by solving a linear equation or an inequality
B1
Obtain critical value \(x = -\dfrac{1}{5}a\) similarly
B2
State final answer \(-\dfrac{1}{5}a < x < \dfrac{5}{3}a\)
B1
Do not condone \(\leqslant\) for \(<\) in the final answer.
Total
4
**Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| **EITHER:** State or imply non-modular inequality $2^2(2x-a)^2 < (x+3a)^2$, or corresponding quadratic equation, or pair of linear equations $2(2x-a) = \pm(x+3a)$ | B1 | |
| Make reasonable attempt at solving a 3-term quadratic, or solve two linear equations for $x$ | M1 | |
| Obtain critical values $x = \dfrac{5}{3}a$ and $x = -\dfrac{1}{5}a$ | A1 | |
| State final answer $-\dfrac{1}{5}a < x < \dfrac{5}{3}a$ | A1 | |
| **OR:** Obtain critical value $x = \dfrac{5}{3}a$ from a graphical method, or by inspection, or by solving a linear equation or an inequality | B1 | |
| Obtain critical value $x = -\dfrac{1}{5}a$ similarly | B2 | |
| State final answer $-\dfrac{1}{5}a < x < \dfrac{5}{3}a$ | B1 | Do not condone $\leqslant$ for $<$ in the final answer. |
| **Total** | **4** | |
1 Find the set of values of $x$ satisfying the inequality $2 | 2 x - a | < | x + 3 a |$, where $a$ is a positive constant. [4]\\
\hfill \mbox{\textit{CAIE P3 2018 Q1 [4]}}