Respuesta :
The inequalities that must be true if m is greater-than-or-equal-to n are given as:
- Option 1: [tex]m \times 2.1 \geq n \times 2.1[/tex]
- Option 2: [tex]m - (-4) \geq n - (-4)[/tex]
- Option 4: [tex]16.5m \geq 16.5n[/tex]
What operations are allowed in inequalities?
Only those operations are allowed which are sure to not modify the inequality between the expressions in the statement.
Some operations that are sure not to modify the inequalities are:
Addition, Subtraction, multiplication by positive real numbers, division by positive real numbers etc.
Example, if 5 > 2, then you can do whatever which keeps the term on the side of 5 bigger than the terms formed on side of 2.
Thus, 5 + 1 > 2 + 1 is correct, (it was addition of 1, a real number).
For the considered case, we're given that:
[tex]m \geq n[/tex]
Checking all the options sequentially, we get:
- Case 1: [tex]m \times 2.1 \geq n \times 2.1[/tex]
It must be true, since the multiplication is done by 2.1, which is > 0
- Case 2: [tex]m - (-4) \geq n - (-4)[/tex]
It must be true since -4 is a real number, and addition or subtraction of any real number (if done on all the sides of the statement) doesn't change the inequality
- Case 3: [tex]m - 3 \geq n + 3[/tex]
From [tex]m \geq n[/tex], addition or subtraction is guaranteed to not change the inequality only if they are of same quantity on all the sides of the inequality. Thus, this inequality is not necessary to be true.
Example, if we take m = 5, n =3, then,
[tex]5 \geq 3[/tex] but [tex]5 - 3=0 \: \rm and \: 3 + 3= 6 \implies 5-3 < 3 + 3[/tex]
- Case 4: [tex]16.5m \geq 16.5n[/tex]
It must be true if given that [tex]m \geq n[/tex] since multiplication of equal positive real number on both the sides of [tex]m \geq n[/tex] was done.
- Case 5: [tex]\dfrac{1}{2} 6m \geq 9m[/tex]
one-half of 6 = 3, thus, this inequality means [tex]3m \geq 9m[/tex]
Dividing both the sides by 3, we get:
[tex]m \geq 3n[/tex]
It is not necessary to be true if [tex]m \geq n[/tex]. Example
[tex]3 \geq 2\\but\\ 3 < 3 \times 2 = 6[/tex]
Thus, the inequalities that must be true if m is greater-than-or-equal-to n are given as:
- Option 1: [tex]m \times 2.1 \geq n \times 2.1[/tex]
- Option 2: [tex]m - (-4) \geq n - (-4)[/tex]
- Option 4: [tex]16.5m \geq 16.5n[/tex]
Learn more about inequality here:
https://brainly.com/question/11901702
