Question

How does one solve $(x - 1)(3x - 4) \geqslant 0$?

Answer 1

Hint: You can solve this inequality simply by setting the values then represent them on the number line then evaluate in each portion whether the given inequality satisfies the given condition or not and lastly select the respective sections.

Complete step by step answer:
We have the following inequality ,
$(x - 1)(3x - 4) \geqslant 0$
We have to set the inequality as
 $
  x - 1 = 0 \\
  and \\
  3x - 4 = 0 \\
 $
So that we get the points and then represent the points on the number line and do further evaluation,
Now the points are as follow ,
$
  x = 1 \\
  and \\
  x = \dfrac{4}{3} \\
 $

Now evaluate that in what portion the given inequalities is positive ,
Between $( - \infty ,1]$ ,
$
  x - 1 \leqslant 0 \\
  and \\
  3x - 4 \leqslant 0 \\
 $
Therefore, $(x - 1)(3x - 4) \geqslant 0$ in this portion .
Now between $[1,\dfrac{4}{3}]$ ,
$
  x - 1 \geqslant 0 \\
  and \\
  3x - 4 \leqslant 0 \\
 $
Therefore, $(x - 1)(3x - 4) \leqslant 0$ in this portion .
Now between $[\dfrac{4}{3},\infty )$ ,
$
  x - 1 \geqslant 0 \\
  and \\
  3x - 4 \geqslant 0 \\
 $
Therefore, $(x - 1)(3x - 4) \geqslant 0$ in this portion .
After evaluating we have our solution i.e.,
$x \in ( - \infty ,1]\bigcup {[\dfrac{4}{3}} ,\infty )$ .

Note: If we multiply positive with positive we get positive , if we multiply positive with negative we get negative ,if we multiply negative with positive we get negative and if we multiply negative with negative we get positive . An inequality is a relationship between two different quantities. The notation $a > b$means that $a$is strictly larger in size than $b$, while the notation $a \geqslant b$ means that $a$ is greater than or equal to $b$.