We will employ a variety of methods, both graphical and algebraic, to solve rational inequalities.

Solve the simple rational inequality

Algebraic Solution

Recall that multiplying both sides of an inequality by a negative value reverses the inequality condition.

Case \(1\): \(x-2 \lt 0 \Rightarrow x \lt 2\)

Since \(x \lt 2\) and \(x \leq 4\), therefore \(x \lt 2\).

Case \(2\): \(x-2 \gt 0 \Rightarrow x \gt 2\)

Since \(x \gt 2\) and \(x \geq 4\), therefore \(x \geq 4\).

Therefore, the solution is \(\left\{ x \mid x \lt 2 \right.\) or \(\left. x \geq 4,\ x \in \mathbb{R} \right\}\).

Solve the simple rational inequality

Algebraic Solution

Therefore, the solution is \(\left\{ x \mid x \lt 2 \right.\) or \(\left. x \geq 4,\ x \in \mathbb{R} \right\}\).

This solution can also be expressed using interval notation, \(x \in (-\infty,2) \cup [4,\infty ),\ x \in \mathbb{R},\) and illustrated using a number line as shown below.

Solve the simple rational inequality

Graphical Solution

Let \(f(x)=\dfrac{6}{x-2}\) and \(g(x)=3\) and graph each function.

To determine the point of intersection of the two functions, set \(f(x)=g(x)\) and solve.

\(g(4)=3\) and \(f(4)=\dfrac{6}{4-2}=3\).

Therefore, the point of intersection is \((4,3)\).

Solve the simple rational inequality

Graphical Solution

Using the graphs and keeping in mind the asymptote, \(x=2\), and the point of intersection, \((4,3)\), we can determine the intervals in which \(f(x) \leq g(x)\).

This gives the solution