Mathematics 9709/22 — May/June 2018

Walkthrough

We start with $|3x - 2| < |x + 5|$. Because an absolute value is always non-negative, both sides are non-negative, so squaring both sides does not change the direction of the inequality. This is the key step: it replaces the awkward modulus signs with an ordinary quadratic inequality.

After squaring, we get $(3x - 2)^2 < (x + 5)^2$. Expanding gives $9x^2 - 12x + 4 < x^2 + 10x + 25$. Subtracting $x^2 + 10x + 25$ from both sides gives $8x^2 - 22x - 21 < 0$.

Next we factorise: $8x^2 - 22x - 21 = (4x + 3)(2x - 7)$. The critical values are where the product is zero: $x = -\frac{3}{4}$ and $x = \frac{7}{2}$.

Since the coefficient of $x^2$ is positive, the quadratic is negative between the two roots. So the inequality holds for $-\frac{3}{4} < x < \frac{7}{2}$.

We could also solve by considering the two linear cases (removing modulus signs piecewise) or by sketching the two modulus graphs; both lead to the same critical values.

Key Takeaways

• Squaring both sides of a modulus inequality is valid when both sides are non-negative.
• A quadratic inequality is solved by factorising, finding critical values, and using the sign of the quadratic.
• The solution to a "less than" quadratic inequality with positive leading coefficient is the interval between the roots.

Common Mistakes

• Squaring without checking both sides are non-negative (here it is safe, but in general it can change the solution set).
• Sign errors when expanding $(3x - 2)^2$ or $(x + 5)^2$.
• Misidentifying the interval: testing a point like $x = 0$ (which gives $| -2 | < |5|$, i.e. $2 < 5$, true) confirms the interval between the roots.
• Including the endpoints: because the original inequality is strict ($<$), the critical values are excluded.

Things to Be Careful About

• The final answer must use strict inequalities: $-\frac{3}{4} < x < \frac{7}{2}$, not $\leq$.
• When factorising, double-check the signs: $(4x + 3)(2x - 7)$ gives the correct expansion.
• The critical values are $-3/4$ and $7/2$; mixing these up or their order would give the wrong interval.