#### Chapter 3 Inequalities in one Variable

**Section 3.1 Inequalities and their Solution Sets**

# 3.1.3 Specific Transformations

The following equivalent transformations are useful if the variable occurs in the denominator of an expression. But they can only be applied under certain restrictions:##### **Info 3.1.9 **

Under the restriction that none of the occurring denominators is zero (the corresponding variable values are never solutions) and the fractions on both sides have the same sign, the reciprocal can be taken on both sides of the inequality while inverting the comparator.

##### **Example 3.1.10 **

For example, the inequality $\frac{1}{2x}\le \frac{1}{3x}$ is equivalent to $2x\ge 3x$ (comparator inverted) as long as $x\ne 0$. The new inequality has the solution set $\text{}]-\infty ;0]$. However, since the value $x=0$ was excluded (and does not belong to the domain of the initial inequality either) the solution set of $\frac{1}{2x}\le \frac{1}{3x}$ is $L=\text{}]-\infty ;0[\text{}$.

##### **Exercise 3.1.11 **

Find the solution sets of the following inequalities.

- $\frac{1}{x}\ge \frac{1}{3}$ has the solution set $L$$\hspace{0.5em}=\hspace{0.5em}$ .

- $\frac{1}{x}<\frac{1}{\sqrt{x}}$ has the solution set $L$$\hspace{0.5em}=\hspace{0.5em}$ .

Please note for the last part of the exercise:

##### **Info 3.1.12 **

Taking the square on both sides of an inequality is not an equivalent transformation and possibly does change the solution set.

For example, $x=-2$ is no solution of $x>\sqrt{x}$, but indeed a solution of ${x}^{2}>x$. However, this transformation can be applied if the case analysis for the transformation is carried out correctly and the domain of the initial inequality is taken into account. This method is described in more detail in the next section.