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The rule of transformation is known to be a basic rule in secondary school mathematics, but is universally applied in most arithmetic problems. So what is the rule of transition in math? State the transition rule? Theory on how to convert addition and multiplication? How is the rule to change the sign in the inequality? In the content of the article below, let’s Tip.edu.vn Find out about the topic of sign conversion rules in the article below!.

Learn about equality and inequality

What is the definition of equality?

In mathematics, equality is understood as a relationship between two quantities. More generally, from two expressions, it is asserted that the two quantities or values ​​are equal, i.e. have the same value, or that both represent the same mathematical object.


The equality between ( a ) and ( b ) is written as ( a=b ) and read as ( a ) as ( b ) , where ( a ) and ( b) ) are called both sides of the equality.

Example 1:

  • ( x=y ) means ( x ) and ( y ) represent the same thing
  • ( (x+1)^{2}=x^{2} + 2x +1 ) means that if ( x ) is any number, the two expressions still have the same value. In this case, it can also be said that both sides of the equality represent the same function.

Example 2: Equalitys

  • 6=8-2
  • ( x^{2} = xx )

The properties of equality

  • Permutation properties: ( a=b ) then ( b=a )
  • Bridging properties: ( a=b ) and ( b=c ) then ( a=c )
  • Properties related to addition and subtraction:
    • ( a=b Rightarrow a+c=b+c )
    • ( a=b Rightarrow ac=bc )
  • Properties related to multiplication and division:
    • ( a=b Rightarrow ac=bc )
    • ( a=b Rightarrow a/c = b/c )

What is the concept of inequality?

  • In mathematics, an inequality is by definition a statement about an ordinal relationship between two objects.
    • The notation ( a< b ) means that ( a ) is less than ( b )
    • The notation ( a>b ) means that ( a ) is greater than ( b )
  • These relations are called strict inequalities; In addition, we have
    • ( a leq b ) means that a is less than or equal to b.
    • ( a geq b ) means that a is greater than or equal to b.
  • Another notation is also used to indicate that one quantity is much larger than another.
    • The notation ( agg b ) means that a is much larger than b.
  • The symbols a and b on both sides of an inequality can be expressions of variables. In the following, we only consider inequalities with variables taking values ​​on the set of real numbers or its subsets.

Comment:

  • If an inequality holds for all values ​​of all the variables present in the inequality, then the inequality is called an absolute or unconditional inequality.
  • If an inequality is only true for certain values ​​of the variables, then for other values ​​it is reversed or no longer true, it is called a conditional inequality.
  • A true inequality remains true if both sides of it are added or subtracted by the same value, or if both sides of it are multiplied or divided by the same positive number.
  • An inequality is often reversed if both sides of it are multiplied or divided by a negative number.

For example: Inequalities:

  • ( 6>4-2 )
  • ( x^{2}+2x^{2}+6>0 )
  • ( -a^{4} -4a^{2} leq 0 )

The theory of inversion rule – sign change rule

State the sign conversion rule

Rule: When converting a term from one side to the other side of any equality, we need to change the sign of that term: the “+” sign into the “-” sign and vice versa, the “-” sign into the “+” sign.

Comment on the rule to change the sign

If ( x=ab ) then according to the conversion rule we have ( x+b=a )

Conversely, if ( x +b = a ) then according to the transformation rule we have ( x=ab )

The foregoing shows that if ( x ) is the difference of ( a ) and ( b ) then ( a ) is the sum of ( x ) and ( b ) . In other words, subtraction is the inverse of addition

For example:

( x+4=y-2 Rightarrow xy=-2-4 Rightarrow xy=-6 [latex]

From this rule we can state some properties of equality:

  • If [latex] a=b ) then ( a+c=b+c )
  • If ( a+c=b+c ) then ( a=b )
  • If ( a=b ) then [/latex] b=a [/latex]

So we have learned the definition and properties of the transformation rule in mathematics. Here, let’s get acquainted with some types of transitional rule exercises.

The sign conversion rule in the inequality

Similar to the equality, we also have the transition rule in the inequality with relatively similar properties.

Rule: When transferring a term from one side of an inequality to the other, we must change the sign of that term: the “+” sign to the “-” sign and the “-” sign to the “+” sign.

The inequality has the same properties as in the equation:

For example:

  • ( a-x+5 geq b Rightarrow ab-x+5 geq 0 )
  • ( x^{3}+2x^{2}-2x< 9 Rightarrow x^{3}+2x^{2}-2x-9 <0 )

Mathematical forms on the sign conversion rule

Form 1: Find the unknown number in an equality

Solution method:

Apply the properties of equality, the bracket rule, and the conversion rule, and then perform the calculation with the known numbers.

Example 1: Find the integer ( x in mathbb{Z} ) known:

  1. ( 6-x= (-5) – 6 )
  2. ( x-3 = 7-9 )

Solution:

  1. ( 6-x= (-5) – 6 )

( Rightarrow 6-x= -11 )

( Rightarrow -x= -11-6 ) (apply property of equality)

( Rightarrow -x= -17 )

( Rightarrow x= 17 )

2. ( x-3 = 7-9 )

( Rightarrow x-3= -2 )

( Rightarrow x= -2+3 ) (apply property of equality)

( Rightarrow x= 1 )

Example 2: (Lesson 63 page 87 textbook)

Find the integer ( x in mathbb{Z} ), knowing that the sum of 3 numbers ( 3; -2; x ) is equal to ( 5 )

Solution:

According to the topic we have:

( 3 + (-2) + x = 5 )

( -2+x=5-3 )

( x=5-3+2 )

( x=4 )

Example 3: Let ( a, b in mathbb{Z} ) . Find the integer ( x in mathbb{Z} ) known:

  1. ( a+x=b )
  2. ( ax=b )

Answer:

  1. ( x=ba )
  2. ( x=ab )

Form 2: Find the number in an equality that contains the sign of absolute value

Solution method:

We need to master the concept of the absolute value of an integer a, which is the distance from point a to point 0 on the number line (in units of length to set up the number line).

  • The absolute value of 0 is 0.
  • The absolute value of a positive integer is itself;
  • The absolute value of a negative integer is its opposite (and a positive integer).
  • Two opposite numbers have the same absolute value.

=> ( left | x right |=a (a in mathbb{N}) ) then ( x=a ) or ( x=-a ) .

For example: (Lesson 62 page 87 textbook)

Find the integer ( a in mathbb{Z} ) known:

  1. ( left | a right |=2 )
  2. ( left | a+2 right |=0 )

Solution:

  1. ( left | a right |=2 ) so ( a=2 ) or ( a=-2 )
  2. ( left | a+2 right |=0 ) so ( a+2=0 ) or ( a=-2 )

Form 3: Calculating algebraic sums

Solution method

Change the position of the term, apply the bracket rule appropriately, and then do the calculation.

Example 1: Calculate

  1. (-54)+(-217)
  2. -45+18
  3. 15-34
  4. 15-27-54
  5. (-15)+75-24

Answer:

  1. -271
  2. -27
  3. -19
  4. -66
  5. 36

Example 2: (Lesson 70 page 88 textbook)

Calculate the following sums reasonably:

  1. ( 3784 + 23 – 3785 – 15 )
  2. ( 21 + 22 + 23 + 24 – 11 – 12 – 13 – 14 )

Solution:

  1. ( 3784 + 23 – 3785 – 15 = (3784 – 3785) + (23 – 15)=-1 + 8=7 )
  2. ( 21 + 22 + 23 +24 – 11 – 12 – 13 – 14 = (21 – 11) + (22 – 12) + (23 – 13) + (24 – 14) = 10 + 10 + 10 + 10 = 40 )

Example 3: (Lesson 71 page 88 textbook)

Quick calculation:

  1. ( – 2001 + (1999 + 2001) ) ;
  2. ( (43 – 863) – (137 – 57) ) .

Solution:

  1. ( – 2001 + (1999 + 2001) = (- 2001 + 2001) + 1999 = 1999 ) ;
  2. ( (43 – 863) – (137 – 57) = 43 – 863 – 137 + 57 = (43 + 57) – (863 + 137) = 100 – 1000 = – 900 ) .

Form 4: The problem returns the addition and subtraction of integers

Solution method:

Based on the problem, deduce to lead to the addition and subtraction of integers

given in advance.

Example 10. (Lesson 68 page 87 textbook)

A football team last year scored 27 goals, while conceding 48 goals again. This year the team scored 39 goals and conceded 24 goals. Calculate that team’s goal-loss difference for each season.

Solution:

To calculate the difference between goals and losses, we have to do integer subtraction. The team’s goal difference – loss last year is 27 – 48 = – 21. Goal difference – loss this year for the team

ball is 39 – 24 = 15.

Answer: Goal difference – loss:

  1. Last year : -21
  2. This year : 15

Example 11. (Lesson 69 page 87 textbook)

The table below shows the highest and lowest temperatures for several cities on a given day. Please write in the right column the number of degrees of difference (the highest temperature minus the lowest temperature) in that day for each city as shown in the table:

learn about sign conversion rules

Solution:

To calculate the number of degrees of difference in a city day, we need to calculate the difference between the highest temperature and the lowest temperature.

Answer: Write in the column in order from top to bottom:

( 9^{circ}C; 6^{circ}C; 14^{circ}C; 10^{circ}C; 12^{circ}C; 7^{circ}C; 13^{circ}C; )

Example 12. (Lesson 72 page 88 textbook)

Quiz: There are 9 cards with numbers and divided into 3 groups as shown in Figure 51 of the textbook.

exercises on the sign conversion rule

Move a card from one group to another such that the sum of the numbers in each group

equally.

Solution:

The sum of the numbers in the three groups is equal to:

[2 + (-1) + (- 3)] + [5 + (- 4) + 3] + [(- 5) + 6 + 9] = (- 2) + 4 + 10 = 12.

After transferring, the sum of the numbers in each group equals : 12 : 3 = 4.

This number is exactly equal to the sum of the numbers in group II. It is necessary to move the cover with number 6 from group III to group I.

Recently, we have become acquainted with some common math forms in the 6th grade program that apply the return rule, here are some exercises for you to practice on your own!.

Some exercises on the rules of self-practice

Multiple choice exercises

Question 1: If ( a+c=b+c ) then:

  1. ( a=b )
  2. ( a
  3. ( a>b )
  4. Both the a, b, c are wrong

Verse 2: Given ( b in mathbb{Z} ) and ( bx=-11 ) . Find ( x ) :

  1. ( -11 – b )
  2. ( -11 + b )
  3. ( b + 11 )
  4. ( -b + 11 )

Question 3: Find ( x ) knowing ( x + 5 = 2 )

  1. ( -3 )
  2. ( -7 )
  3. ( 3 )
  4. ( 7 )

Question 4: Which of the following integers ( x in mathbb{Z} ) satisfy ( x-7=20 ) ?

  1. ( x = 27 )
  2. ( x = 13 )
  3. ( x = -12 )
  4. ( x = -27 )

Question 5: How many integers ( x in mathbb{Z} ) are there such that ( x +13= 445 )?

  1. 0
  2. first
  3. 2
  4. 3

Essay exercises

Lesson 1: Find the integer ( x in mathbb{Z} ) known:

( 7 – (19+14) = x + (17-25) )

Lesson 2: Find the integer ( x in mathbb{Z} ) known:

  1. ( left | x-5 right |=4 )
  2. ( left | x+7 right |=0 )

Lesson 3: Given integers ( x, y in mathbb{Z} ). Let’s prove that:

  1. If ( xy>0 ) then ( x>y )
  2. If ( x>y ) then ( xy>0 )

Lesson 4: It is proved that: The distance between two points ( a, b ) on the number line (( a, b in mathbb{Z} )) is equal to ( left | ab right |=0 ) or ( left | ba right |=0 ) . Find the distance between the points ( a ) and ( b ) on the number line when:

  1. ( a=-5 ) ; ( b=7 )
  2. ( a=-7 ) ; ( b=-4 )
  3. ( a=12 ) ; ( b=6 )

Lesson 5: Find the integer ( x in mathbb{Z} ) knowing that ( x-8 ) is the smallest two-digit negative integer:

Lesson 6: Prove that: ( left | ab right |= left | ba right | )

Lesson 7: A kite flies up to a height of 15m, then lowers to 5m, then rises to 7m, lowers to 6m and then rises to 9m in the wind. What is the final height of the kite?

With the above detailed article, Tip.edu.vn hopes to help you understand the most basic issues of equality, inequality in general and the sign conversion rule in particular cases. Applying these properties through some basic and advanced exercises will help you review your knowledge effectively. If you have any questions or contributions to the article content on the topic “Rules for converting signs”, don’t forget to leave them in the comment section below. Good luck with your studies!.

See the details of the lecture below:


(Source: www.youtube.com)

See more:

  • Topics on subtraction and division: Basic math forms and Exercises
  • What is the definition of prime number? Superprime number? 2 prime numbers together?

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