Standard +0.8 This requires systematic case analysis of two modulus expressions with different critical points (x=1/3 and x=-1), solving multiple inequalities across three regions, and combining solutions. More demanding than routine single-modulus problems but standard technique for P3 level.
State or imply non-modular inequality \(2^2(3x-1)^2 < (x+1)^2\), or corresponding quadratic equation, or pair of linear equations
B1
Form and solve a 3-term quadratic, or solve two linear equations for \(x\)
M1
e.g. \(35x^2 - 26x + 3 = 0\)
Obtain critical values \(x = \dfrac{3}{5}\) and \(x = \dfrac{1}{7}\)
A1
Allow 0.143 or better
State final answer \(\dfrac{1}{7} < x < \dfrac{3}{5}\)
A1
Exact values required. Accept \(x > \dfrac{1}{7}\) and \(x < \dfrac{3}{5}\). Do not condone \(\leqslant\) for \(<\) in the final answer. Fractions need not be in lowest terms.
Alternative method for Question 1:
Answer
Marks
Guidance
Answer
Marks
Guidance
Obtain critical value \(x = \dfrac{3}{5}\) from a graphical method, or by solving a linear equation or linear inequality
B1
Obtain critical value \(x = \dfrac{1}{7}\) similarly
B2
Allow 0.143 or better
State final answer \(\dfrac{1}{7} < x < \dfrac{3}{5}\)
B1
OE. Exact values required. Accept \(x > \dfrac{1}{7}\) and \(x < \dfrac{3}{5}\). Do not condone \(\leqslant\) for \(<\) in the final answer. Fractions need not be in lowest terms.
Total
4
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply non-modular inequality $2^2(3x-1)^2 < (x+1)^2$, or corresponding quadratic equation, or pair of linear equations | **B1** | |
| Form and solve a 3-term quadratic, or solve two linear equations for $x$ | **M1** | e.g. $35x^2 - 26x + 3 = 0$ |
| Obtain critical values $x = \dfrac{3}{5}$ and $x = \dfrac{1}{7}$ | **A1** | Allow 0.143 or better |
| State final answer $\dfrac{1}{7} < x < \dfrac{3}{5}$ | **A1** | Exact values required. Accept $x > \dfrac{1}{7}$ **and** $x < \dfrac{3}{5}$. Do not condone $\leqslant$ for $<$ in the final answer. Fractions need not be in lowest terms. |
**Alternative method for Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain critical value $x = \dfrac{3}{5}$ from a graphical method, or by solving a linear equation or linear inequality | **B1** | |
| Obtain critical value $x = \dfrac{1}{7}$ similarly | **B2** | Allow 0.143 or better |
| State final answer $\dfrac{1}{7} < x < \dfrac{3}{5}$ | **B1** | OE. Exact values required. Accept $x > \dfrac{1}{7}$ **and** $x < \dfrac{3}{5}$. Do not condone $\leqslant$ for $<$ in the final answer. Fractions need not be in lowest terms. |
| **Total** | **4** | |