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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

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