What is the largest modulus. The absolute value of a number. Complete Lessons - Knowledge Hypermarket

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A module is a mathematical concept that is passed in the sixth grade. By itself, the numerical module is nothing complicated, it is one of the simplest topics in elementary mathematics. But if you accidentally skip the study of the desired paragraph, then you may encounter a misunderstanding of the topic. Therefore, we recall what exactly is called a module, how to find it for different numbers, and what is this concept in essence.

Module in terms of geometry

Looking ahead, let's try to immediately understand what a module is in practice - it will be easier to grasp its meaning. Let's draw a coordinate line on a sheet of paper, take zero as a reference point, and on the right and on left side at the same distance put some two points - for example, 5 and -5.

The module will be considered exactly the actual distance to zero from -5 and from 5. Obviously, this distance will be exactly the same. Therefore, in both cases, the modulus will be equal to the number "5" - and it does not matter what sign is in front of the original number that we are considering.

How to find the modulus of a number?

Now that we have a visual representation of what a module is, it will be easier to understand the wording from the textbook. It says that the modulus of a certain number is this number itself, if it is positive, the number opposite to the original number if it is negative, and zero if we are looking for the modulus for zero.

This can be formulated in another way - the modulus of any number will be the number itself in absolute terms, that is, without taking into account the sign. The module is written like this - on both sides of the right number vertical lines are placed, for example, the module for the number "5" will be equal to "5", and it will be written as |5|.

From all that we have said above, we can deduce a few strict rules for modules.

• Can modulus be negative? Not! Modulus can only be positive. Even if we are talking about a negative number, for example, -7, then its modulus will be equal to |7| - a number opposite to the original.
• For zero, the modulus will always be zero. Another thing is also true - zero can be a module only if it is calculated for the number zero, and in no other way.
• If you need to find the module for an expression like a * b, that is, the module of the product, then you can first find the module a, then the module b, and multiply them by each other.
• The same goes for division - if we need to divide y by z and find the modulus of the resulting number, then we can take the modulus of y and divide it by the modulus of z. The result will be the same.

Module numbers n is the number of unit segments from the origin to point n. And it does not matter in which direction this distance will be counted - to the right or left of zero.

Instruction

• Module numbers also called the absolute value of this numbers. It is indicated by short vertical lines drawn to the left and right of numbers. For example, the module numbers 15 is written as follows: |15|.
• Remember that the modulus can only be a positive number or zero. Modulus of positive numbers equal to the number itself. The modulus of zero is zero. That is, for any numbers n greater than or equal to zero, the following formula |n| = n. For example, |15| = 15, that is, the modulus numbers 15 equals 15.
• Modulo negative numbers will be the same number, but with opposite sign. That is, for any numbers n, which is less than zero, the formula |n| = -n. For example, |-28| = 28. Module numbers-28 is equal to 28th.
• You can find modules not only for integers, but also for fractional numbers. And the same rules apply to fractional numbers. For example, |0.25| = 25, that is, the modulus numbers 0.25 will be equal to 0.25. A |-¾| = ¾, that is, the modulus numbers-¾ will be equal to ¾.
• When working with modules, it is useful to know that the modules of opposite numbers are always equal to each other, that is, |n| =|-n|. This is the main property of modules. For example, |10| = |-10|. Module numbers 10 equals 10, just like a modulus numbers-ten. Moreover, |a - b| = |b - a|, since the distance from point a to point b and the distance from b to a are equal to each other. For example, |25 - 5| = |5 - 25|, i.e. |20| = |- 20|.

Opposite numbers are numbers that differ from each other only in sign. Expression -a means it is a number opposite number a.

For example, 7 and - 7;
41 and - 41, etc.

The number 0 is the opposite of itself!

That is, to show opposite numbers in mathematics use the sign « – ».

Attributing the sign "-" in front of a positive number 5 , we get a negative number – 5 .

Attributing the sign "-" before negative number – 5 , we get the opposite positive number 5 , that is - (-5) = 5.

- (-a) \u003d a

On a coordinate line, points with opposite coordinates are located at the same distance from the origin.

AO=OC
BO=OD

The absolute value of a number

The absolute value of a number- this is the distance (in unit segments) from the origin to the point that depicts this number on the coordinate line.

Points A (- 4) and B (4) are 4 unit segments away from the origin, and the numbers - 4 and 4 have the same modules equal to 4.

The modulus of a is denoted by | a |

Since the modulus is the distance, and the distance cannot be negative, then modulus of a number cannot be a negative number!!!

The modulus of a positive number and zero is the same number, and the modulus of a negative number is its opposite number:

| a | = a if a ≥ 0 (if a is a non-negative number)

| a | = - a, if a< 0 (если а – отрицательное число)

conclusions

Number module properties:

1. The modulus of a number cannot be negative. The modulus of a number is always either a positive number or equal to 0.

1. Opposite numbers have equal modules.

| – a | = | a | = a

Example, | – 12 | = | 12 | = 12

Solving equations (examples)

1.-x=7
instead of -x and 7 we write their opposite numbers using the "-" sign
–(– x) = – 7
we use the rule that – (–a) = a we get
x = -7

2. – x = – 10
–(– x) = –(– 10)
x=10

3. x = -(- 32)
x=32

4. | x | = 4
x=4 or x=-4
Answer: 4; - four

5. | x | = 0
x=0
Answer: 0

6. | y | = - 8
the modulus cannot be a negative number, which means that this equation has no solution
Answer: no roots

7. | – x | = 12
remember the second property of the module, that| - a| = |a| = a, then
| x | = 12
x=12 or x=-12
Answer: 12; - 12

8. | y | – 2 = 12
similar equations are solved as simple equations, only taking into account the module
| y | = 12 + 2
| y | = 14
y=14 or y=-14
Answer: 14; - fourteen

9.10 – 2| x | = 4
2| x | = 10 - 4
2| x | = 6
| x | = 6:2
| x | = 3
x = 3 or x = – 3
Answer: 3; – 3

That is, when solving equations containing a module, we get three types of answers:
two roots (if the modulo sign is a positive number), one root (if modulo 0)
no roots (if the module sign is a negative number).

Solution of the simplest inequalities containing the modulus

In the 5th grade, we solved examples with the simplest inequalities. Linear inequalities are strict and non-strict.
Strict inequalities are inequalities with greater than (>) or less than (<).
x > a; x< a;
Non-strict inequalities are inequalities with signs greater than or equal to (≥) or less than or equal to (≤).
x ≥ a; x ≤ a.

Examples

1. Find all natural values ​​of x for which the inequality x is correct< 9

Solution.
This inequality will be correct for such values ​​of x: 1; 2; 3; four; 5; 6; 7; eight.
Answer: x \u003d (1; 2; 3; 4; 5; 6; 7; 8) - natural solutions this inequality.

Note:
The number 0 is not a solution to this inequality, since 0 is not a natural number;
The number 9 is not a solution to this inequality, since this inequality is strict, that is, x is strictly less than 9 and cannot be equal to 9.

2. a satisfies the inequality a> 12?

Solution.
Since the inequality is strict, the number 13 is the smallest natural value of a that satisfies this inequality.
Answer: 13

3. What is the smallest natural value a satisfies the inequality a ≥ 12?

Solution.
Since the inequality is not strict, the number 12 is the smallest natural value of a that satisfies this inequality.
Answer: 12.

4. < x < 9

Solution.
The inequality is double (read as "x is greater than 2, but less than 9"), strict, therefore 3; four; 5; 6; 7; 8 - natural solutions of this double inequality.
Answer: x = (3; 4; 5; 6; 7; 8)

5. Find all natural values ​​of x for which inequality 2 is correct< x ≤ 9.

Solution.
3; four; 5; 6; 7; eight; 9 - natural solutions of this double inequality.
Answer: x = (3; 4; 5; 6; 7; 8; 9)

6. Find all integers that satisfy the inequality | x |< 5.

Solution.
| x |< 5 (читаем как «расстояние от начала отсчёта до точки изображающей х меньше 5»).
Inequality | x |< 5 эквивалентно (can also be written) –5 < x < 5. Неравенство двойное, строгое, поэтому данное неравенство будет правильным при таких значениях x: –4; –3; –2; –1; 0; 1; 2; 3; 4.
Answer: x = (-4; -3; -2; -1; 0; 1; 2; 3; 4)

7. Find all integers that satisfy the inequality | x | ≤ 5.

Solution.
Inequality | x | ≤ 5 is equivalent to –5 ≤ x ≤ 5. The inequality is double, not strict, so the numbers –5 and 5 will be included in the set of numbers for which this inequality is correct. Thus, this inequality will be correct for such values ​​of x: –5; -four; -3; –2; -one; 0; one; 2; 3; four; 5.
Answer: x = (-5; -4; -3; -2; -1; 0; 1; 2; 3; 4; 5)

8. Find all integers that satisfy the inequality | x | > 2 and label them on the coordinate line.

Solution.
Inequality | x | > 2 is equivalent to x< – 2 или x >2. Denote on the coordinate line the points whose coordinates satisfy the given inequality

Since the inequality is strict, then the numbers - 2 and 2 are not included in the set of integers for which this inequality will be correct. And on the coordinate line, these points are denoted as an unshaded point.

Answer: x \u003d (... -5; -4; -3; 3; 4; 5 ...)

9. Find all integers that satisfy the inequality | x | ≥ 2 and label them on the coordinate line.

Solution.
Inequality | x | ≥ 2 is equivalent to x ≤ – 2 or x ≥ 2. Denote on the coordinate line the points whose coordinates satisfy the given inequality

Since the inequality is not strict, then the numbers - 2 and 2 are included in the set of integers for which this inequality will be correct. And on the coordinate line, these points are denoted as a filled point.

Answer: x \u003d (... -5; -4; -3; -2; 2; 3; 4; 5 ...)

10. Find all integers that satisfy inequality 1< | x | ≤ 3 и обозначте их на координатной прямой.

Solution.
Consider first the left side of the inequality. It means that the distance from the origin to the points is less than 1. Consider the right side of the inequality: the distance from the origin to the same points is less than or equal to 3.
Let's construct these points on the coordinate line:

1 and -1 are not included in the set of integers that satisfy the inequality, because the inequality is strict.
3 and -3 are included in the set of integers that satisfy the inequality, because the inequality is not strict.

Answer: x = (-3; -2; 2; 3)