Review practice · Section 1 of 3

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

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

Show that $\dfrac{2x^2+2x-8}{x^2+x-6}-1=\dfrac{x^2+x-2}{x^2+x-6}$.1 mark

Hence, without using a calculator, solve the inequality $\dfrac{2x^2+2x-8}{x^2+x-6}\leq1$.4 marks

Deduce the solution to the inequality $\dfrac{4x^2+2x-4}{2x^2+x-3}\leq1$.2 marks

(i) Without using a calculator, solve the inequality $\dfrac{5x-1}{x^2+x-6}\leq1$.3 marks

(ii) Hence solve the inequality $1+\dfrac{1-5\ln x}{(\ln x)^2+\ln x-6}\geq0$.3 marks

Without the use of a calculator, solve the inequality $\dfrac1x+1\leq x(1+x)$.3 marks

Hence, solve $1-e^{2x}+e^{-x}-e^x\leq0$.2 marks

Without using a calculator, solve the inequality $\dfrac{x^3+3x^2+2x+1}{x^3-2x^2}\geq1$.4 marks

Given that $x$ is real, prove algebraically that $x^2-2x+3$ is always positive.1 mark

Hence, without using a graphing calculator, find the range of values of $x$ that satisfy $\dfrac{x+4}{x^2-2x+3}\geq\dfrac3{2-x}$.5 marks