Practice section 5 of 7

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

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

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

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

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

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

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

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

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

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]

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

Hence solve $\dfrac{3|x|}{2x^2-3|x|+1}>-1$. [3]