Additional Mathematics 4037/22 — October/November 2013

Walkthrough

The inequality $x^2 < 6 - 5x$ must first be rewritten with everything on one side, giving $x^2 + 5x - 6 < 0$. This is essential because we can only reason about the sign of a single expression.

Next we factorise $x^2 + 5x - 6$. We need two numbers multiplying to $-6$ and adding to $+5$; these are $+6$ and $-1$, so the factorisation is $(x + 6)(x - 1)$.

The critical values are where $(x + 6)(x - 1) = 0$, namely $x = -6$ and $x = 1$. These are the only places where the sign of the expression can change.

Since the coefficient of $x^2$ is positive, the graph of $y = x^2 + 5x - 6$ is a U-shaped parabola. It lies below the $x$-axis (i.e. the expression is negative) strictly between the two roots. Hence the solution set is $-6 < x < 1$.

Key Takeaways

• Always move all terms to one side before solving a quadratic inequality.
• Factorise (or use the formula) to locate the critical values.
• For a positive-coefficient quadratic, the expression is negative between the roots and positive outside them.

Common Mistakes

• Writing the answer as $x > -6$ OR $x < 1$ — the mark scheme explicitly rejects 'OR' or a comma; both conditions must hold simultaneously ('AND').
• Solving $x^2 < 6 - 5x$ by taking square roots or splitting it incorrectly instead of collecting terms first.
• Choosing the wrong region (outside the roots) by forgetting the parabola's shape.
• Sign errors when moving $-5x$: it becomes $+5x$ on the left-hand side.

Things to Be Careful About

• The final answer must be written as a strict double inequality $-6 < x < 1$ (or as two conditions joined by AND); the mark scheme says 'Mark final answer', so a correct method with a wrongly stated final interval loses the last mark.
• The inequalities are strict ($<$), so the endpoints $-6$ and $1$ are not included — do not write $\leq$.