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$\frac{x}{3}+\frac{4}{5} < \frac{5}{3}$

                                      Subtract  $\frac45$  from both sides of the inequality.

$\frac{x}{3}+\frac{4}{5}{\color{blue}-\frac45} < \frac{5}{3}{\color{blue}-\frac45}$

$\frac{x}{3} < \frac{5}{3}-\frac45$

                                      Get a common denominator on the right side.

$\frac{x}{3} < \frac{5}{3}\cdot\frac55-\frac45\cdot\frac33$

$\frac{x}{3} < \frac{25}{15}-\frac{12}{15}$

                                      Combine the fractions on the right side.

$\frac{x}{3} < \frac{13}{15}$

                                      Multiply both sides by  3 , a positive number.

$\frac{x}{3}{\color{blue}\cdot3} < \frac{13}{15}{\color{blue}\cdot3}$

$x < \frac{39}{15}$

                                      And  $\frac{39}{15}$  =  2.6

$x <2.6$

What is the largest integer  x  such that  x < 2.6   ?

What is the largest integer less than  2.6 ?

The largest integer less than  2.6  is  2 , so

x  =  2