College Algebra (10th Edition)

Solution set: $\{-1,3\}$
$\sqrt{2x+3}-\sqrt{x+1}=1$ ...Add $\sqrt{x+1}$ to both sides $\sqrt{2x+3}=1+\sqrt{x+1}$ ...Square both sides $(\sqrt{2x+3})^{2}=(1+\sqrt{x+1})^{2}$ ...Apply $(a+b)^{2}=a^{2}+2ab+b^{2}$ $2x +3==1+2\sqrt{x+1}+(x+1)$ $2x +3=x+2\sqrt{x+1}+2$ $2x+3-x=x+2\sqrt{x+1}+2-x$ $x+3=2\sqrt{x+1}+2$ $x+1=2\sqrt{x+1}$ ...Square both sides $x^{2}+2x+1=4x+4$ $x^{2}-2x-3=0$ $(x-3)(x+1)=0$ $x=3,$ test: $\sqrt{2(3)+3}-\sqrt{3+1}=3-2=1$ x=3... is a solution. $x=-1,$ test: $\sqrt{2(-1)+3}-\sqrt{-1+1}=1-0=1$ x=-1... is a solution. Solution set: $\{-1,3\}$