Question

How do you solve \[(x - 1)(3x - 4) \geqslant 0\] using a sign chart?

Answer 1

Hint: Sign chart is made for the roots of the inequality. Determine the critical points and check the sign of the function in the ranges between and beyond these critical points. Then, find the answer as per the sign given in the equation.

Complete step-by-step solution:
Let \[f(x) = \] \[(x - 1)(3x - 4) \geqslant 0\]
Therefore, the roots of the inequality are \[1\] and $\dfrac{4}{3}$ as for these values the inequality returns a zero value when these values are substituted in the inequality.
Roots can be found by the following method-
The expression has two parts in multiplication, therefore, even if one of them is zero, the entire equation will become zero.
If we consider \[(x - 1)\] , this expression will become zero if the value of \[x\] is \[1\] . You could also find when an expression will become zero by equating it to zero and finding the value of the variable.
\[(x - 1) = 0\]
\[ \Rightarrow x = 1\]
Similarly, equate \[(3x - 4)\] to zero and find for what value it would become zero.
\[(3x - 4) = 0\]
\[ \Rightarrow 3x = 4\]
\[ \Rightarrow x = \dfrac{4}{3}\]
Therefore, for these two values, the inequality becomes zero, hence, \[1\] and $\dfrac{4}{3}$ are the roots of this inequality.
The sign chart for this inequality will be

\[x\]	\[ - \infty \]	\[1\]	\[\dfrac{4}{3}\]	\[ - \infty \]
\[(x - 1)\]	\[ - \]	\[0\]	\[ + \]	\[ + \]
\[(3x - 4)\]	\[ - \]	\[ - \]	\[0\]	\[ + \]
\[f(x)\]	\[ + \]	\[0\]	\[0\]	\[ + \]

Therefore, \[f(x) \geqslant 0,\] when \[x \in \left( { - \infty ,1} \right] \cup \left[ {\dfrac{4}{3}, + \infty } \right)\] .

Note: The polynomial should be written in the correct form. Correct form refers that it should be written in the descending order of the powers of the variable. It must be greater than, less than, greater than equal to or less than equal to.
Next, one should find the critical points of the inequality. Critical points refer to those points whose value when substituted in the inequality, returns a value, zero. These critical values can be referred to as the roots of the inequality. Once, the critical points are obtained, check the sign of the function in the ranges beyond and between these critical points and tabulate them. This all can be tabulated and shown in a sign chart. Depending upon the inequality, decide the range of the answer.