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What is the largest integer $x$ such that $\frac{x}{3}+\frac{4}{5} < \frac{5}{3}$?

May 18, 2019

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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

May 18, 2019
edited by hectictar  May 18, 2019