"Basic Properties of Inequalities" PPT Courseware 2

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"Basic Properties of Inequalities" PPT Courseware 2

Question 1: The temperature of lightning is about 28,000°C, which is 4.5 times higher than the surface temperature of the sun. Assuming that the surface temperature of the sun is t°C, what relationship should t satisfy?

4.5t<28000

Question 2: Use appropriate symbols to represent the following relationships:

(1) The sum of 2x and 3 is not greater than -6;

(2) The difference between 5 times x and 1 is less than 3 times x;

(3) The difference between a and b is a negative number.

Definition of inequality

An expression that expresses an inequality relationship using an inequality sign (＞, ≥, ＜, ≤ or ≠) is called an inequality

For example, 4.5t<28000, 2x+3≤6, a-b<0, etc. are all inequalities.

Note: Not greater than, that is, less than or equal to, is expressed by "≤";

Not less than, that is, greater than or equal to, is represented by "≥".

Basic property of equation 1:

If you add (or subtract) the same integer to both sides of the equation, the equation still holds

If a=b, then a±c=b±c

Basic property 2 of equation:

If both sides of the equation are multiplied or divided (the divisor cannot be 0) by the same number, the equation still holds.

If a=b, then ac=bc or a/c=b/c (c≠0),

Do inequalities have similar properties?

if 7 > 3

Then 7+5 ____ 3+ 5 , 7 -5____3-5

If -1<3,

Then -1+2____3+2, -1- 4____3 - 4

Can you summarize the pattern?

Basic property of inequality 1: If the same number or the same integer is added (or subtracted) to both sides of the inequality, the direction of the inequality sign remains unchanged.

That is: if a>b, then a±c>b±c.

Basic property of inequality 2: Both sides of the inequality are multiplied (or divided) by the same positive number, and the direction of the inequality sign remains unchanged.

If a>b, c>0, then ac>bc (or a/c>b/c)

Basic property of inequality 3: If both sides of the inequality are multiplied (or divided) by the same negative number, the direction of the inequality sign changes.

If a>b, c<0, then ac

Symmetry of inequalities: if a>b, then b

Isotropic transitivity of inequalities: if a>b, b>c, then a>c

Example 1: Suppose a>b, fill in the blanks with “＜” or “＞” and answer orally which basic property of the inequality is based on.

(1) a - 3____b - 3;

(2) a÷3____b÷3

(3) 0.1a____0.1b;

(4) -4a____-4b

(5) 2a+3____2b+3;

(6) (m2+1) a ____ (m2+1)b (m is a constant)

Example 2: Determine whether the derivation of the following questions is correct? Why

(1) Because 7.5>5.7, so -7.5<-5.7;

(2) Because a+8>4, so a>-4;

(3) Because 4a>4b, so a>b;

(4) Because -1>-2, so -a-1>-a-2;

(5) Because 3>2, so 3a>2a.

practise

1. If m>n, determine whether the following inequality is correct:

(1) m-7

(2) 3m<3n ( )

(3)-5m>-5n ( )

(4) m/9>n/9 ( )

(5) m+5≥n+5 ( )

2. Fill in the blanks

(1) If x-5>4, then both sides can be _____ to get x>9

(2) If you add 9 to both sides of -7<8, you can get _____

(3) If we add a+2 to both sides of 5>-2, we get _____

(4) If you multiply both sides of -3>-4 by 7, you can get _____

(5) If you multiply both sides of 8>0 by 8, you get _____

Summary of this lesson:

Today we are learning the five basic properties of inequalities:

Basic properties of inequalities 1:

If a > b, then a±c > b±c. That is to say, the same number (or formula) is added (or subtracted) to both sides of the inequality, and the direction of the inequality sign remains unchanged.

Basic properties of inequalities 2:

If a > b, c > 0, then ac > bc (or a/c > b/c) means that both sides of the inequality are multiplied (or divided) by the same positive number, and the direction of the inequality sign remains unchanged.

Basic property of inequalities 3:

If a>b, c<0, then ac

Symmetry of inequalities: if a>b, then b

Inequality is transitive: if a>b, b>c, then a>c

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Update Time: 2024-11-16

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