Mathematics 9709/32 — May/June 2021

Walkthrough

1. Remove the modulus: Both $|2x - 1|$ and $3|x + 1|$ are non-negative for all real $x$. Because squaring preserves the inequality direction for non-negative quantities, we can safely square both sides to obtain $(2x - 1)^2 < 9(x + 1)^2$.
2. Expand and rearrange: Expanding both sides gives $4x^2 - 4x + 1 < 9x^2 + 18x + 9$. Moving all terms to one side yields the quadratic inequality $5x^2 + 22x + 8 > 0$.
3. Find critical values: Solve the corresponding equation $5x^2 + 22x + 8 = 0$ using the quadratic formula. The discriminant is $22^2 - 4(5)(8) = 324$, giving roots $x = -4$ and $x = -2/5$.
4. Interpret the solution: The quadratic $5x^2 + 22x + 8$ is a parabola opening upwards. It is positive (greater than zero) outside the interval between its roots. Thus, the solution is $x < -4$ or $x > -2/5$.

Key Takeaways

• Squaring both sides is a valid and efficient method for solving modulus inequalities when both sides are non-negative.
• The solution to a quadratic inequality $ax^2 + bx + c > 0$ with $a > 0$ is the region outside the roots.
• Always verify that the final answer uses the correct inequality signs ($<$ vs $\leq$) as dictated by the original problem.

Common Mistakes

• Incorrect inequality direction: When rearranging terms, students may accidentally flip the inequality sign.
• Wrong inequality signs in final answer: Using $\leq$ or $\geq$ instead of $<$ or $>$, which does not match the original strict inequality.
• Swapping the solution regions: Writing $-4 < x < -2/5$ instead of the correct regions outside the roots. This happens when confusing the solution to $> 0$ with the solution to $< 0$.
• Forgetting to check non-negativity: Squaring both sides is only valid when both sides are non-negative. Here, modulus expressions are always non-negative, so it is safe.

Things to Be Careful About

• The original inequality is strict ($<$), so the final answer must also be strict ($<$ and $>$). Do not include equality.
• The solution consists of two disjoint intervals; use "or" to connect them, not "and". Writing $-2/5 < x < -4$ is mathematically impossible and scores zero.
• Equivalent bracket notation such as $x \in (-\infty, -4) \cup (-0.4, \infty)$ is acceptable, but ensure the union symbol and brackets are used correctly.