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Manipulation of inequalities

Solve the fol...

Question

Solve the following inequality:
$(x^{2} + 3 x + 1) (x^{2} + 3 x - 3) ⩾ 5$

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Solution

$(x^{2} + 3 x + 1) ((x^{2} + 3 x + 3) \geq 5$
Let $x^{2} + 3 x$ be t
$(t + 1) (t - 3) \geq 5 t^{2} - 2 t - 3 - 5 \geq 0 t^{2} - 2 t - 8 \geq 0 (t - 4) (t + 2) \geq 0 ∴ (x^{2} + 3 x - 4) ((x^{2} + 3 x + 2) \geq 0 (x + 4) (x - 1) (x + 2) (x + 1) \geq 0 x \in (- \infty, - 4) \cup (- 2, - 1) \cup (1, \infty)$

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Manipulation of inequalities

Standard XII Mathematics

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