In the article we will consider solution of inequalities... We will tell you in an accessible way about how to construct a solution to inequalities, with clear examples!
Before considering the solution of inequalities using examples, let's understand the basic concepts.
General information about inequalities
Inequality is called an expression in which functions are connected by relation signs>,. Inequalities are both numerical and alphabetic.
Inequalities with two signs of the relationship are called double, with three - triple, etc. For instance:
a (x)> b (x),
a (x) a (x) b (x),
a (x) b (x).
a (x) Inequalities containing the sign> or or are not strict.
Solving inequality is any value of the change at which this inequality is true.
"Solve inequality"means that it is necessary to find many of all its solutions. There are various methods for solving inequalities... For solutions to inequality use the number line, which is infinite. For instance, solution of inequality x> 3 is an interval from 3 to +, and the number 3 is not included in this interval, therefore a point on a straight line is denoted by an empty circle, since the inequality is strict. 
+
The answer will be: x (3; +).
The value x = 3 is not included in the solution set, so the parenthesis is round. The infinity sign is always surrounded by a parenthesis. The sign means "belonging".
Consider how to solve inequalities using another signed example:
x 2
-
+
The value x = 2 is included in the set of solutions, therefore the bracket is square and a point on the line is denoted by a filled circle.
The answer will be: x. The decision set graph is shown below. 
Double inequalities
When two inequalities are connected by a word and, or, then it is formed double inequality... Double inequality like
-3
and 2x + 5 ≤ 7
called connected because it uses and... Writing -3 Double inequalities can be solved using the principles of addition and multiplication of inequalities.
Example 2 Solve -3 Solution We have
The set of solutions (x | x ≤ -1 or x> 3). We can also write a solution using spacing notation and a symbol for amalgamations or inclusions of both sets: (-∞ -1] (3, ∞). The graph of the solution set is shown below. 
To test, draw y 1 = 2x - 5, y 2 = -7, and y 3 = 1. Note that for (x | x ≤ -1 or x> 3), y 1 ≤ y 2 or y 1> y 3. 
Inequalities with absolute value (modulus)
Inequalities sometimes contain modules. The following properties are used to solve them.
For a> 0 and an algebraic expression x:
| x | | x | > a is equivalent to x or x> a.
Similar statements for | x | ≤ a and | x | ≥ a.
For instance,
| x | | y | ≥ 1 is equivalent to y ≤ -1 or y ≥ 1;
and | 2x + 3 | ≤ 4 is equivalent to -4 ≤ 2x + 3 ≤ 4.
Example 4 Solve each of the following inequalities. Plot the set of solutions.
a) | 3x + 2 | b) | 5 - 2x | ≥ 1
Solution
a) | 3x + 2 |
b) | 5 - 2x | ≥ 1
The solution set is (x | x ≤ 2 or x ≥ 3), or (-∞, 2])
