Hint
Solve each inequality separately, draw their intervals on a number line, then take the overlap.
Worked solution
(a)
\(2x^2-5x-3=(2x+1)(x-3)\), so the roots are \(\displaystyle - \frac{1}{2}\) and 3.
Answer: \(\displaystyle x=-\frac12,3\).
(b)
The leading coefficient is positive, so the expression is nonpositive between its roots, including both endpoints.
Answer: \(\displaystyle -\frac12\le x\le3\).
(c)
The second inequality gives x<−2 or x>2. Its intersection with \(\displaystyle [- \frac{1}{2}, 3]\) is (2,3]. The endpoint 2 is excluded; 3 is allowed.
Answer: \(2<x\le3\).
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 2 marks
- 1 mark: Correct factorisation or quadratic-formula method.
- 1 mark: Both roots \(\displaystyle - \frac{1}{2}\) and 3.
Alternative methods: Any valid exact-root method is accepted.
Common error: A sign error in the smaller root.
(b) · 2 marks
- 1 mark: Choose the interval between the roots using a sign argument.
- 1 mark: Include both endpoints.
Alternative methods: A sign table or upward-parabola argument is accepted.
Common error: Selecting the exterior intervals.
(c) · 2 marks
- 1 mark: Obtain x<−2 or x>2 for the second condition.
- 1 mark: Intersect correctly, with 2 excluded and 3 included.
Alternative methods: A clear number-line overlap is sufficient.
Common error: Taking a union instead of the intersection.
Total: 6 marks.