First slide

Introduction to linear inequalities

Question

$Number of positive integers satisfying \frac{(x + 2) (x^{2} - 2 x + 1)}{- 4 + 3 x - x^{2}} ≥ 0 is$

Moderate

A

0
B

1
C

2
D

3

Solution

$\begin{array}{l} \frac{(x + 2) (x^{2} - 2 x + 1)}{- x^{2} + 3 x - 4} \geq 0 \\ \Rightarrow \frac{(x + 2) (x - 1)^{2}}{x^{2} - 3 x + 4} \leq 0 \\ but x^{2} - 3 x + 4 > 0 \forall x \in R \sin c e ∆ = 9 - 4 \cdot 1 \cdot 4 = - 7 < 0 \\ ∴ (x + 2) (x - 1)^{2} \leq 0 \\ \Rightarrow x \in (- \infty, - 2] \cup {1} \\ t h e r e f o r e o n l y 1 i s t h e p o s i t i v e i n t e g r a l s o l u t i o n \end{array}$

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