1/(24*x)>-1 inequation

Given the inequality:
$$\frac{1}{24 x} > -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{1}{24 x} = -1$$
Solve:
Given the equation:
$$\frac{1}{24 x} = -1$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case

a1 = 1

b1 = 1

a2 = 1

b2 = -24*x

so we get the equation
$$- 24 x = 1$$
$$- 24 x = 1$$
Divide both parts of the equation by -24

x = 1 / (-24)

We get the answer: x = -1/24
$$x_{1} = - \frac{1}{24}$$
$$x_{1} = - \frac{1}{24}$$
This roots
$$x_{1} = - \frac{1}{24}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + - \frac{1}{24}$$
=
$$- \frac{17}{120}$$
substitute to the expression
$$\frac{1}{24 x} > -1$$
$$\frac{1}{\left(- \frac{17}{120}\right) 24} > -1$$

-5/17 > -1

the solution of our inequality is:
$$x < - \frac{1}{24}$$

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