Practice section 4 of 7

It is given that $f(x)=\dfrac{x-5}{2x-3}$. Using an algebraic method, solve the inequality $f(x)\leq\dfrac23$.

Hence find the exact range of values of $x$ for which

(i) $f(\ln x)\leq\dfrac23$,

(ii) $f\left(\left|x+\dfrac12\right|\right)\leq\dfrac23$.

It is given that $y=\dfrac{x^2+x+1}{x-1}$, $x\in\mathbb R$, $x\ne1$.

Without using a calculator, find the set of values that $y$ can take.

(a) Show that if $x^2>k(x+1)$ for all real $x$, then $-4<k<0$.

(b) Find the range of values of $k$ for which $x^2+2kx+15-2k$ is always positive for all real $x$. Deduce the range of values of $k$ for which $k(x^2+2kx+15-2k)>0$ for all real $x$.

Given that $a$ is a positive real number, solve the inequality $\dfrac{x^2-x+2}{x^2+(1-a)x-a}\leq0$, leaving your answer in terms of $a$.

Without using a calculator, solve $\dfrac{3x-1}{1-2x^2}\geq1$.

Hence find the solution of $\dfrac{3\sin\theta-1}{\cos2\theta}\geq1$, where $-\dfrac\pi2\leq\theta\leq\dfrac\pi2$.

Without the use of a calculator, solve $\dfrac6{x-1}<x$.

Hence, solve $\dfrac6{|x|-1}<|x|$.