Let's look at examples of how to solve the system linear inequalities.
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To solve a system, you need each of its constituent inequalities. Only the decision was made to record not separately, but together, combining them curly brace.
In each of the inequalities of the system, we transfer the unknowns to one side, the known ones to the other with opposite sign:
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After simplification, both sides of the inequality must be divided by the number in front of X. We divide the first inequality by positive number, so the inequality sign does not change. We divide the second inequality by a negative number, so the inequality sign must be reversed:
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We mark the solution to the inequalities on the number lines:
In response, we write down the intersection of the solutions, that is, the part where there is shading on both lines.
Answer: x∈[-2;1).
In the first inequality, let's get rid of the fraction. To do this, multiply both sides term by term by the smallest common denominator 2. When multiplied by a positive number, the inequality sign does not change.
In the second inequality we open the brackets. The product of the sum and the difference of two expressions is equal to the difference of the squares of these expressions. On the right side is the square of the difference between the two expressions.
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We move the unknowns to one side, the known ones to the other with the opposite sign and simplify:
We divide both sides of the inequality by the number in front of X. In the first inequality, we divide by a negative number, so the sign of the inequality is reversed. In the second, we divide by a positive number, the inequality sign does not change:
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Both inequalities have a “less than” sign (it doesn’t matter that one sign is strictly “less than”, the other is loose, “less than or equal”). We can not mark both solutions, but use the “ “ rule. The smaller one is 1, therefore the system reduces to the inequality
We mark its solution on the number line:
Answer: x∈(-∞;1].
Opening the parentheses. In the first inequality - . It is equal to the sum of the cubes of these expressions.
In the second, the product of the sum and the difference of two expressions, which is equal to the difference of squares. Since here there is a minus sign in front of the brackets, it is better to open them in two stages: first use the formula, and only then open the brackets, changing the sign of each term to the opposite.
We move the unknowns in one direction, the knowns in the other with the opposite sign:
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Both are greater than signs. Using the “more than more” rule, we reduce the system of inequalities to one inequality. The larger of the two numbers is 5, therefore,
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We mark the solution to the inequality on the number line and write down the answer: 
Answer: x∈(5;∞).
Since in algebra systems of linear inequalities occur not only as independent tasks, but also during the solution various kinds equations, inequalities, etc., it is important to master this topic in time.
Next time we will look at examples of solving systems of linear inequalities in special cases when one of the inequalities has no solutions or its solution is any number.
Category: |In the article we will consider solving inequalities. We will tell you clearly about how to construct a solution to inequalities, with clear examples!
Before we look at solving inequalities using examples, let’s understand the basic concepts.
General information about inequalities
Inequality is an expression in which functions are connected by relation signs >, . Inequalities can be both numerical and literal.
Inequalities with two signs of the ratio 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.
Solving the inequality is any value of the variable for which this inequality will be true.
"Solve inequality" means that we need to find the set of all its solutions. There are different methods for solving inequalities. For inequality solutions They use the number line, which is infinite. For example, solution to inequality x > 3 is the interval from 3 to +, and the number 3 is not included in this interval, therefore the point on the line is denoted by an empty circle, because 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 always stands out parenthesis. The sign means "belonging."
Let's look at how to solve inequalities using another example with a sign:
x 2
-
+
The value x=2 is included in the set of solutions, so the bracket is square and the point on the line is indicated by a filled circle.
The answer will be: x)
