Inequalities are the expressions of the form:

f(x) ≥ 0

where instead of ≥ sign can be written ≤ sign or < and >.

To solve the inequality above, one should find the set of values of variable x such as f(x) greater than or equal to 0.

Look at the plot of arbitrary function f(x):

From the plot above, intervals of the values of the variable х such as f(x) ≥ 0 (filled with light green color), can immediately be written out:

f (x) ≥ 0
<=>
{ x є (−∞;
x_{1}] U
[x_{2};
x_{3}] U
[x_{4};
+∞)
}

From the plot above follows that the function changes its sign in the points of intersection of х axis. Therefore, to solve any inequality, one should first of all find the values of x such as function f(x) equals to zero, i.e solve the equaltion f(x) = 0.

Found set of values of variable
x_{i}
(i.e roots of equaltion
f(x)=0) splits abscissa axis into the intervals where value of function keeps its sign (greater than or less than zero).

To solve corresponding inequality one should determine the sign of the function in each interval and choose the intervals which satisfy the inequality. To determine function sign in any interval
(x_{i};
x_{j}) one should substitute value of
x
in the expression
f(x)
by any value
x_{k} є (x_{i};
x_{j}).

Our online calculator, based on Wolfram Aplha LLC system, capable to solve many kinds of different inequalities with step by step solution.

Inequalities online solver

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