Hint
Both expressions are real. Solve both signed linear equations.
Worked solution
(a)
\(3 x + 11 = 5 x + 23\) and \(3 x + 11 = - 5 x - 23\) give the displayed two roots.
Answer: \(\displaystyle \left\{-6, - \frac{17}{4}\right\}\)
(b)
\(\left(3 x + 11\right)^{2} - \left(5 x + 23\right)^{2} \leq 0\) has negative leading coefficient, so it holds outside and at the two roots.
Answer: \(\displaystyle \left(-\infty, -6\right] \cup \left[- \frac{17}{4}, \infty\right)\)
(c)
Subtract smaller from larger.
\(\displaystyle \frac{7}{4}\)
Answer: \(\displaystyle \frac{7}{4}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Solve \(\displaystyle U=V\) and \(\displaystyle U=-V\) and check both roots.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Solving only \(\displaystyle U=V\) loses a branch.
(b) · 2 marks
- 1 mark: Compare \(\displaystyle U^{2}\) and \(\displaystyle V^{2}\) and use a sign chart with included equality endpoints.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not keep a duplicated solution twice.
(c) · 2 marks
- 1 mark: Subtract smaller from larger.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Solving only \(\displaystyle U=V\) loses a branch.
Total: 6 marks.