Need to show this inequality for an economic proof When would the following inequality be true (Any relationship between $b$ and $d$ ?) $$\frac{ax -b}{cx - d} \geq \frac{a - b}{c - d}$$ where, $0 < x < 1$ and $a

Need to show this inequality for an economic proof When would the following inequality be true (Any relationship between $b$ and $d$ ?) $$\frac{ax -b}{cx - d} \geq \frac{a - b}{c - d}$$ where, $0 < x < 1$ and $a < c$ I need to prove this inequality to get to a economic proof. The true value of $x$ would be around ~ 0.999. Thanks in advance

Not necessarily. Assume that $ad-bc < 0$ and $cx > d$. Then

$$x < 1 \implies (ad-bc)x > (ad-bc) \implies -ad - bcx > -bc - adx$$

$$\implies acx - ad - bcx + bd > acx -bc-adx + bd \implies (cx-d)(a-b) > (ax-b)(c-d)$$

$$\frac{a-b}{c-d} > \frac{ax-b}{cx-d}$$