ANSWER
[tex]x = 6[/tex]
EXPLANATION
The given equation is
[tex] \sqrt{3x + 7} = x - 1[/tex]
We square both sides of the equation to obtain,
[tex](\sqrt{3x + 7} ) ^{2} =( x - 1)^{2} [/tex]
This implies that,
[tex]3x + 7 = {x}^{2} - 2x + 1[/tex]
Rewrite in standard quadratic equation form.
[tex] {x}^{2} - 2x - 3x+ 1 - 7 = 0[/tex]
[tex]{x}^{2} -5x - 6= 0[/tex]
Factor
[tex](x - 6)(x + 1) = 0[/tex]
This implies that,
[tex]x = 6 \: x = - 1[/tex]
We check for extraneous solution by substituting each x-value into the original equation.
When x=-1,
[tex]\sqrt{3( - 1)+ 7} = - 1- 1[/tex]
[tex]\sqrt{4} = - 2[/tex]
2=-2....False
Hence x=-1 is an extraneous solution.
When x=6,
[tex]\sqrt{3( 6)+ 7} = 6- 1[/tex]
[tex]\sqrt{25} = 5[/tex]
5=5 is True.
Hence x=6 is the only solution.
