Solution of exponential inequalities. Solving inequalities

In the article we will consider solution of inequalities. Let's talk plainly about how to build a solution to inequalities with clear examples!

Before considering the solution of inequalities with examples, let's deal with the basic concepts.

Introduction to inequalities

inequality is called an expression in which functions are connected by relation signs >, . Inequalities can be both numerical and alphabetic.
Inequalities with two relation signs are called double, with three - triple, etc. For example:
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.
Inequality solution is any value of the variable for which this inequality is true.
"Solve the inequality" means that you need to find the set of all its solutions. There are various methods for solving inequalities. For inequality solutions use a number line that is infinite. For example, solving the inequality x > 3 is an interval from 3 to +, and the number 3 is not included in this interval, so the point on the line is denoted by an empty circle, because the inequality is strict.
+
The answer will be: x (3; +).
The value x=3 is not included in the set of solutions, so the parenthesis is round. The infinity sign is always enclosed in a parenthesis. The sign means "belonging".
Consider how to solve inequalities using another example with the sign:
x2
-+
The value x=2 is included in the set of solutions, so the square bracket and the point on the line is denoted by a filled circle.
The answer will be: x . The solution 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. Record -3 Double inequalities can be solved using the principles of addition and multiplication of inequalities.

Example 2 Solve -3 Solution We have

Set of solutions (x|x ≤ -1 or x > 3). We can also write the solution using the spacing notation and the symbol for associations or inclusions of both sets: (-∞ -1] (3, ∞). The graph of the set of solutions 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 example,
|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|