IB Mathematics Analysis and Approaches Question Type 1: Simple Equation Proofs Exercises

Show that $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{b}-\frac{1}{a}}=\frac{a+b}{a-b}$, for $a,b\ne 0$ and $a\ne b$.

Show that $\frac{1}{x+h}-\frac{1}{x}=-\frac{h}{x(x+h)}$, for $x\ne 0$ and $x+h\ne 0$.

Show that ${\left(1+\frac{1}{x}\right)}^2-{\left(1-\frac{1}{x}\right)}^2=\frac{4}{x}$, for $x\ne 0$.

Prove that $\sqrt{a}-\sqrt{b}=\frac{a-b}{\sqrt{a}+\sqrt{b}}$, for distinct positive real numbers $a,b$.

Prove that $\frac{x^2+x+1}{x+2}=x-1+\frac{3}{x+2}$, for $x\ne -2$.

Prove that $\frac{1}{x-1}-\frac{1}{x+1}=\frac{2}{x^2-1}$, for $x\ne \pm 1$.

Verify that $\frac{1+x}{1-x}-\frac{1-x}{1+x}=\frac{4x}{1-x^2}$, for $x\ne \pm 1$.

Show that $\frac{1}{\sqrt{x}+\sqrt{y}}+\frac{1}{\sqrt{x}-\sqrt{y}}=\frac{2\sqrt{x}}{x-y}$, for $x\ne y$, $x,y>0$.

Prove that $\frac{x^3-y^3}{x-y}=x^2+xy+y^2$, for $x\ne y$.

Prove that $\frac{(n+1)!}{n!}=n+1$, for integer $n\ge 0$.

Prove that $\frac{1}{m+1}+\frac{1}{m^2+m}=\frac{1}{m}$, where $m\ne 0,-1$.

Show that $\frac{a^2-b^2}{a-b}=a+b$, for $a\ne b$.