Question

How do you solve \[\left( {x - 1} \right)\left( {3x - 4} \right) \geqslant 0\]?

Answer 1

Hint: These questions can be solved by using the fact that when the inequality is in the form \[\left( {x - a} \right)\left( {x - b} \right) \geqslant 0\] and \[a < b\], then \[x \leqslant a\]and \[x \geqslant b\], we can took all constant terms to one side and all terms containing \[x\] to the other sides and now substituting the values and using the sign chart we get the required result.

Complete step-by-step answer:
A linear inequality is an inequality in one variable that can be written in one of the following forms where \[a\] and \[b\] are real numbers and \[a \ne 0\],
\[ax + b < 0;ax + b > 0;ax + b \geqslant 0;ax + b \leqslant 0\].
Now given inequality is \[\left( {x - 1} \right)\left( {3x - 4} \right) \geqslant 0\],
We know that if the inequality is in the form \[\left( {x - a} \right)\left( {x - b} \right) \geqslant 0\]and\[a < b\], then \[x \leqslant a\] and \[x \geqslant b\] so,
Equating the terms as in the formula we get,
\[ \Rightarrow x - 1 \leqslant 0\] and \[3x - 4 \geqslant 0\],
Move the constant term to the other side we get,
$ \Rightarrow x \leqslant 1$ and $ \Rightarrow 3x \geqslant 4$,
Now divide 3 to both sides on the second term we get,
$ \Rightarrow x \leqslant 1$ and $\dfrac{{3x}}{3} \geqslant \dfrac{4}{3}$,
Now simplifying we get,
$ \Rightarrow x \leqslant 1$ and $x \geqslant \dfrac{4}{3}$,
Now using the sign chart,

\[x\]	\[ - \infty \]	1	\[\dfrac{4}{3}\]	\[\infty \]
\[x - 1\]	\[ - \]	0	+	+
\[3x - 4\]	\[ - \]	\[ - \]	0	+
Given function	+	0	0	+

So, the solution for the given inequality can be written as \[\left( { - \infty ,1} \right] \cup \left[ {\dfrac{4}{3},\infty } \right)\]

The solution of the given inequality \[\left( {x - 1} \right)\left( {3x - 4} \right) \geqslant 0\]is\[\left( { - \infty ,1} \right] \cup \left[ {\dfrac{4}{3},\infty } \right)\].

Note:
There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities.