Question

How do you solve $ \dfrac{4}{3}r - 3 < r + \dfrac{2}{3} - \dfrac{1}{3}r $ ?

Answer 1

Hint: In this question we are given an inequality, as the inequality contains an unknown variable quantity “r” and numerical values, and “r” and the numerical values are linked via arithmetic operations like addition, subtraction, multiplication and division so it is an algebraic expression. We can easily find the value of “r” by rearranging the equation such that the terms containing “r” are present on one side of the equation and the other side contains all other terms. Then we can find the value of “r” by applying the given arithmetic operations.

Complete step-by-step answer:
We are given that \[\dfrac{4}{3}r - 3 < r + \dfrac{2}{3} - \dfrac{1}{3}r\]
To find the value of s, we will take $ s\,and\,\dfrac{1}{3}s $ to the left-hand side and 3 to the right-hand side so that all the terms containing “r” remain on one side and the constant terms are present on the other side –
\[
  \dfrac{4}{3}r + \dfrac{1}{3}r - r < \dfrac{2}{3} + 3 \\
   \Rightarrow \dfrac{{5 - 3}}{3}r < \dfrac{{11}}{3} \\
   \Rightarrow \dfrac{2}{3}r < \dfrac{{11}}{3} \;
 \]
Now, we will take $ \dfrac{2}{3} $ to the right-hand side –
 $
  r < \dfrac{{11}}{3} \times \dfrac{3}{2} \\
   \Rightarrow r < \dfrac{{11}}{2}\,or\,r < 5.5 \;
  $
Hence, when \[\dfrac{4}{3}r - 3 < r + \dfrac{2}{3} - \dfrac{1}{3}r\] , we get $ r < \dfrac{{11}}{2} $ .
So, the correct answer is “ $ r < \dfrac{{11}}{2} $ ”.

Note: In this question, we are given that \[\dfrac{4}{3}r - 3\] is smaller than $ r + \dfrac{2}{3} - \dfrac{1}{3}r $ . The given equation is a linear equation as the degree of the equation is 1. We know that we need an “n” number of equations to find the value of “n” unknown variables. In the given algebraic expression; we have 1 unknown quantity and exactly one equation so the value of “r” is obtained easily. In the expression $ r < \dfrac{{11}}{3} \times \dfrac{3}{2} $ , we see that 3 is present in both the numerator and the denominator so we cancel it out and get the value of “r” in simplified form.