Mathematics 9709/33 — October/November 2020

Walkthrough

Start by viewing the inequality graphically. The right-hand side $2|x-3|$ is a V-shaped graph with its vertex at $x=3$, $y=0$. The left-hand side $2-5x$ is a straight line with slope $-5$. The inequality asks for the values of $x$ where the straight line lies above the V.

Because the modulus changes sign at $x=3$, split the problem.

For $x<3$, the modulus graph has equation $y = 6 - 2x$. Equating this to $2-5x$ gives

so $x = -\frac{4}{3}$, which lies in the branch $x<3$.

For $x\geq 3$, the modulus graph is $y = 2x - 6$. Equating to $2-5x$ gives $x = \frac{8}{7}$, but $\frac{8}{7} < 3$, so this intersection is not on the correct branch and is rejected.

To decide the side of the inequality, test $x=0$: $2-5(0)=2$ and $2|0-3|=6$. Since $2>6$ is false, the region containing $x=0$ does not work. Therefore the solution is $x<-\frac{4}{3}$.

Key Takeaways

The question tests sketching a modulus graph and a line to solve an inequality. The critical idea is that the boundary is the first intersection, and the direction of the inequality is determined by a test point. Splitting the modulus at its vertex is an essential algebraic skill.

Common Mistakes

• Omitting the sketch; the mark scheme explicitly awards a mark for a recognisable sketch of $y=2|x-3|$ and the line $y=2-5x$.
• Solving only one branch of the modulus, or accepting the intersection $x=\frac{8}{7}$ without checking it belongs to $x\geq 3$.
• Giving the final answer as a decimal such as $x < -1.33$ rather than the exact value $-\frac{4}{3}$.
• Using $\leq$ instead of the strict $<$ required by the question.

Things to Be Careful About

• The final answer must be exact and strict: $x < -\frac{4}{3}$, not $x \leq -\frac{4}{3}$ and not $x < -1.33$.
• Always check that a calculated intersection lies within the branch of the modulus being considered.
• When testing a point, choose one not on the boundary. Testing $x=0$ gives a quick check that the interval to the right of $-\frac{4}{3}$ is not part of the solution.