"Basic Properties of Inequalities" PPT Courseware 3

"Basic Properties of Inequalities" PPT Courseware 3

Knowledge review

(1) If the same integer is added (or subtracted) to both sides of the equation, the result is still an equation.

If a=b, then a+c=b+c (or a-c=b-c)

(2) If both sides of the equation are multiplied (or divided) by the same integer (the divisor cannot be zero), the result is still an equation.

If a=b, then ac=bc (or a/c=b/c, c≠0)

Fill in the blanks with ">" or "<"

5___ -3

(1)5+3___ -3+3

(2)5-3 ___ -3-3

(3)5×3___-3×3

(4)5×(-3)___-3×(-3)

As a result, the direction of the inequality sign remains the same or changes?

Properties of Inequalities 1 If the same integer is added (or subtracted) to both sides of the inequality, the direction of the inequality sign remains unchanged.

Property 2 of inequalities: If both sides of an inequality are multiplied (or divided) by the same positive number, the direction of the inequality sign remains unchanged.

Property 3 of inequalities: When both sides of an inequality are multiplied (or divided) by the same negative number, the direction of the inequality sign changes.

practice

1. If x+5>4, then _______ both sides to get x>-1

2. Add 9 to both sides of -7 < 8 to get _______.

3. Subtract 6 from both sides of 5＞-2 to get _______.

4. Multiply both sides of -3>-4 by 7 to get _______.

5. Divide both sides of -8<0 by 8 to get _______.

knowledge application

1. If -3<0, then -3+1<1 ( )

2. If -3×2>-5 ×2, then -3<-5 ( )

3. If a

4. If -6a<-6 b, then a< b ( )

5. If a>b, then -a < -b ( )

6. If -2x >0, then x > 0 ( )

7. If -2<1, then -2a < a ( )

8. If a >0, then 3a > 2a ( )

think about it

a is any rational number. Try comparing the sizes of 5a and 3a.

Is this solution correct? If it is correct, tell which basic property of the inequality it is based on; if it is incorrect, please explain the reason.

Answer: This solution is incorrect because we do not know the value range of the letter a. If a<0, then 5a<3a; if a=0, then 5a=3a.

Example 1 According to the basic properties of inequalities, transform the following inequalities into the form x＜a or x＞a:

(1) x -7>2 (2) 6 x <5 x -1

(3) 4x-5＜5x (4)-1/4x＜-1

Solution (1) According to property 1 of the inequality, add 2 to both sides to get:

x－7＋7＞2+7

That is x > 9

(2) According to property 1 of the inequality, subtract 5 x from both sides to get:

6 x -5 x < (5 x -1) -5 x

That is, x <-1

Focus of this section

(1) Master the three properties of inequalities, especially property 3;

The three properties of inequalities are:

① Add (or subtract) the same integer to both sides of the inequality, and the direction of the inequality sign remains unchanged;

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

③ When both sides of the inequality are multiplied (or divided) by the same negative number, the direction of the inequality sign needs to change;

(2) Be able to correctly apply properties to transform inequalities;

Precautions

When both sides of the inequality are multiplied (or divided) by the same number, be sure to see whether it is a positive or negative number; letters without a given range should be discussed on a case-by-case basis.

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