Solving fractional rational inequalities online calculator. Solving inequalities with modulus. Linear inequalities. Solution, examples
Inequality it is an expression with, ≤, or ≥. For example, 3x - 5 Solving an inequality means finding all values \u200b\u200bof the variables for which this inequality is true. Each of these numbers is a solution to the inequality, and the set of all such solutions is its many solutions... Inequalities that have the same set of solutions are called equivalent inequalities.
Linear inequalities
The principles for solving inequalities are similar to those for solving equations.Principles for solving inequalities
For any real numbers a, b, and c:
The principle of adding inequalities: If a The multiplication principle for inequalities: If a 0 is true, then ac If a bc is also true.
Similar statements also apply for a ≤ b.
When both sides of an inequality are multiplied by a negative number, the sign of the inequality needs to be reversed.
The first level inequalities, as in example 1 (below), are called linear inequalities.
Example 1 Solve each of the following inequalities. Then depict many solutions.
a) 3x - 5 b) 13 - 7x ≥ 10x - 4
Decision
Any number less than 11/5 is a solution.
The set of solutions is (x | x
To check, we can plot y 1 \u003d 3x - 5 and y 2 \u003d 6 - 2x. Then it is clear from this that for x 
The solution set is (x | x ≤ 1), or (-∞, 1]. The solution set graph is shown below. 
Double inequalities
When two inequalities are connected by a word and, orthen it is formed double inequality... Double inequality like
-3
and 2x + 5 ≤ 7
called connectedbecause it uses and... Writing -3 Double inequalities can be solved using the principles of addition and multiplication of inequalities.
Example 2 Solve -3 Decision We have
The set of solutions (x | x ≤ -1 or x\u003e 3). We can also write a solution using spacing notation and a symbol for associations or inclusions of both sets: (-∞ -1] (3, ∞). The graph of the solution set is shown below. 
To test, draw y 1 \u003d 2x - 5, y 2 \u003d -7, and y 3 \u003d 1. Note that for (x | x ≤ -1 or x\u003e 3), y 1 ≤ y 2 or y 1\u003e y 3. 
Inequalities with absolute value (modulus)
Inequalities sometimes contain modules. The following properties are used to solve them.
For a\u003e 0 and an algebraic expression x:
| x | | x | \u003e a is equivalent to x or x\u003e 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
Decision
a) | 3x + 2 |
b) | 5 - 2x | ≥ 1
The solution set is (x | x ≤ 2 or x ≥ 3), or (-∞, 2] The following example uses such a bracket.
Let's write down the answer: x ≥ -0,5 at intervals:
x ∈ [-0.5; + ∞)
Read: x belongs to the interval from minus 0.5, including, to plus infinity.
Infinity can never turn on. It is not a number, it is a symbol. Therefore, in such records, infinity is always adjacent to a parenthesis.
This form of notation is convenient for complex answers consisting of several intervals. But - just for the final answers. In intermediate results, where a further solution is expected, it is better to use the usual form, in the form of a simple inequality. We will deal with this in the relevant topics.
Popular jobs with inequalities.
The linear inequalities themselves are simple. Therefore, often, tasks become more complicated. So, to think it was necessary. It’s not very pleasant if you’re not used to it.) But useful. I will show examples of such tasks. Not for you to learn them, it's unnecessary. And in order not to be afraid when meeting with such examples. Think a little - and everything is simple!)
1. Find any two solutions to the inequality 3x - 3< 0
If it is not very clear what to do, remember the main rule of mathematics:
If you don't know what is needed, do what you can!)
x < 1
So what? Nothing special. What are they asking us? We are asked to find two specific numbers that solve an inequality. Those. fit the answer. Two any numbers. Actually, this is embarrassing.) A couple of 0 and 0.5 are suitable. A pair of -3 and -8. Yes, these couples are endless! What is the correct answer ?!
The answer is: everything! Any pair of numbers, each less than one, would be the correct answer. Write what you want. Let's go further.
2. Solve the inequality:
4x - 3 ≠ 0
Quests in this form are rare. But, as auxiliary inequalities, when finding the ODZ, for example, or when finding the domain of definition of a function, they are often encountered. This linear inequality can be solved as an ordinary linear equation. Only everywhere, except for the "\u003d" sign ( equally) put the sign " ≠ " (not equal). So you will come to the answer, with an inequality sign:
x ≠ 0,75
In more complex examples, it is better to do it differently. Make inequality equal. Like this:
4x - 3 = 0
Calmly solve it, as taught, and get the answer:
x \u003d 0.75
The main thing, at the very end, when writing down the final answer, is not to forget that we have found the X, which gives equality. And we need - inequality. Therefore, we just don't need this X.) And we need to write it down with the correct icon:
x ≠ 0,75
This approach results in fewer errors. Those who solve the equations automatically. And for those who do not solve the equations, inequalities, in fact, are useless ...) Another example of a popular task:
3. Find the smallest integer solution to the inequality:
3 (x - 1) < 5x + 9
First, we just solve the inequality. We open the brackets, transfer them, give similar ones ... We get:
x > - 6
Wrong !? Did they follow the signs !? And behind the signs of members, and behind the sign of inequality ...
Thinking again. We need to find a specific number that matches both the answer and the condition "smallest integer".If it doesn't immediately dawn, you can just take any number and estimate. Is two more than minus six? Sure! Is there a suitable smaller number? Of course. For example, zero is greater than -6. And even less? We need the smallest possible! Minus three is more than minus six! You can already grasp the pattern and stop going through numbers stupidly, right?)
We take a number closer to -6. For example, -5. The answer is executed, -5 > - 6. Can you find another number, less than -5, but more than -6? You can, for example, -5.5 ... Stop! We are told wholedecision! Doesn't roll -5.5! Minus six? Uh-uh! The inequality is strict, minus 6 is not less than minus 6!
So the correct answer is -5.
I hope everything is clear when choosing a value from the general solution. Another example:
4. Solve inequality:
7 < 3x + 1 < 13
How! This expression is called triple inequality. Strictly speaking, this is a shorthand notation for the system of inequalities. But you still have to solve such triple inequalities in some tasks ... It is solved without any systems. For the same identical transformations.
It is necessary to simplify, to bring this inequality to a pure xx. But ... What is where to transfer !? Now is the time to remember that the shift left-right is abbreviated form first identical transformation.
And the full form sounds like this: You can add / subtract any number or expression to both sides of the equation (inequality).
There are three parts here. So we will apply identical transformations to all three parts!
So, let's get rid of the 1 in the middle of the inequality. Subtract one from the entire middle part. To prevent inequality from changing, we subtract 1 from the remaining two parts. Like this:
7 -1< 3x + 1-1 < 13-1
6 < 3x < 12
Already better, right?) It remains to divide all three parts into three:
2 < x < 4
That's all. This is the answer. X can be any number from two (not including) to four (not including). This answer is also written at intervals, such records will be in square inequalities. There they are the most common thing.
At the end of the lesson, I will repeat the most important thing. Success in solving linear inequalities depends on the ability to transform and simplify linear equations. If at the same time watch out for the sign of inequality, there will be no problems. Which is what I wish you. No problem.)
If you like this site ...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Instant validation testing. Learning - with interest!)
you can get acquainted with functions and derivatives.
