For test point x = –2, the inequality is true.
For test point x = 0, the inequality is false.
For test point [tex]x=\frac{4}{5}[/tex], the inequality is true.
For test point x = 2, the inequality is false.
Solution:
Given expression is the inequality.
[tex]$\frac{5 x-3}{(x-1)(x+1)} \leq 0[/tex]
Substitute x = –2 in the inequality.
[tex]$\frac{5 (-2)-3}{(-2-1)(-2+1) }=\frac{-13}{(-3)(-1) }[/tex]
[tex]$=\frac{-13}{3 }[/tex]
< 0
For test point x = –2, the inequality is true.
Substitute x = 0 in the inequality.
[tex]$\frac{5 (0)-3}{(0-1)(0+1) }=\frac{-3}{(-1)(1) }[/tex]
[tex]$=3[/tex]
> 0
For test point x = 0, the inequality is false.
Substitute [tex]x=\frac{4}{5}[/tex] in the inequality.
[tex]$\frac{5 (\frac{4}{5} )-3}{((\frac{4}{5} )-1)((\frac{4}{5} )+1) }=\frac{1}{(-\frac{1}{5} )(\frac{9}{5} ) }[/tex]
[tex]$=\frac{-25}{9 }[/tex]
< 0
For test point [tex]x=\frac{4}{5}[/tex], the inequality is true.
Substitute x = 2 in the inequality.
[tex]$\frac{5 (2)-3}{(2-1)(2+1) }=\frac{7}{(1)(3) }[/tex]
[tex]$=\frac{7}{3 }[/tex]
> 0
For test point x = 2, the inequality is false.
Hence,
For test point x = –2, the inequality is true.
For test point x = 0, the inequality is false.
For test point [tex]x=\frac{4}{5}[/tex], the inequality is true.
For test point x = 2, the inequality is false.
