Inequalities

Solve $(2x-18)/(x+3)>x-6$ algebraically. Hence solve $(\ln(x^2)-18)/(\ln x+3)>\ln x-6$.

(a) Use graphs to solve $\sqrt x\ge2x$. (b) Solve $(x-1)(x-3)/(x-2)^2<0$.

(a) Solve $(x^2-6)/(2x)>1$ exactly. Hence solve $(x^2-6)/(2|x|)\ge1$. (b) Prove algebraically that $x^2-4x+9>0$ for all real x. Hence solve $(x^3+2x^2+x+14)/(x^2+5)>x+1$.

(a) Without a calculator solve $(x+5)/(x-1)>9/x$. Hence solve $(e^x+5)/(e^x-1)>9e^{-x}$. (b) For $y=(1+kx)/(x^2-1)$ with real x≠±1 and $-1\le k\le1$, find the range in terms of k.

Solve $x|x-2|\ge1$ exactly. Hence solve $(x+1)|x-1|\ge1$.

Show that $\tan x+\cot x>4$ for $0<x<\pi/2$ simplifies to $0<\sin2x<1/2$. Hence solve it exactly in terms of pi.

Show $x^2-x+1>0$ for every real x. Hence find all a such that $(x^2+ax-2)/(x^2-x+1)<2$ for every real x.

Without a graphing calculator solve $x^2/(x-1)\le4x\le8$.

Show $x^2(x+2)^2-1=(x+1)^2(x+1+\sqrt2)(x+1-\sqrt2)$. Hence solve $1/(x+2)^2>x^2$ without a calculator. Deduce the inputs satisfying $1/(e^x+2)^2>e^{2x}$.

Without a graphing calculator solve $x\le15/(x+2)$. Hence solve $1/x^2\ge(x^2+2)/15$.

(i) Without a calculator solve $4/(x+3)\le x$. (ii) Hence solve $4x/(1+3x)\le1/x$.

(i) Find the exact roots of $|(4x-5)/(x-2)|=3x$. (ii) Sketch that absolute-value curve and y=3x together. Hence solve $|(5-4x)/(3x-6)|>x$ exactly.

(i) For a<0 and b<−1 solve $b|x-a|=x-ab$. (ii) Given a<b<0, (a) state the condition on b for a negative real root; (b) under that condition sketch y=x−ab and y=b|x−a| together, solve $b|x-a|\ge x-ab$, and deduce the inputs satisfying $-b|e^x+a|\le e^x+ab$.

(i) Sketch $y=|(1/3)^{2x}-18|$, finding exact intercepts. (ii) Without a calculator, solve $|(1/3)^{2x}-18|\le9$ with full working, giving the exact interval or intervals in their simplest form.