Respuesta :
Answer:
A) ... reciprocal ... can change the direction
Step-by-step explanation:
You want to know the multiplicative inverse property of inequality.
Multiplicative inverse
The term "multiplicative inverse" is another name for "reciprocal". This immediately rules out choices B, C, D.
The appropriate property is A, the reciprocal can change the direction of the inequality.
Cases
The multiplicative inverse of a number gives 1 when multiplied by the original number. On a number line, this maps numbers greater than 1 to numbers between 0 and 1, and vice versa. Numbers keep their relationship with respect to 1: numbers farther from 1 remain farther from 1.
Here is an example:
3 > 2 and the reciprocal relationship is 1/3 < 1/2 (inequality reversed)
When both numbers are negative, the same sort of relationship holds. In this case the invariant point is -1.
-3 < -2 and the reciprocal relationship is -1/3 > -1/2 (reversed)
However, when the numbers have different signs, the inequality is not reversed:
-3 < 2 and the reciprocal relationship is -1/3 < 1/2 (not reversed)
Since the signs of the numbers don't change when their reciprocal is taken, the relationship between a positive number and a negative number remains unchanged: the positive number is still greater.
__
Additional comment
The square root and logarithm functions have positive slope everywhere, so the direction of the inequality is unaffected.
An absolute value function has a negative slope on one side of its vertex and a positive slope on the other side. Depending on the particular values involved, an absolute value function may or may not reverse an inequality. When solving inequalities involving absolute value functions, the effect of the function in each domain must be considered separately.
Final answer:
The Multiplicative Inverse Property of Inequality concerns taking the reciprocal of both sides of an inequality and reversing the inequality's direction, applicable only when both sides are positive. Option A is correct, as the other options (B, C, D) do not relate to this property.
Explanation:
The Multiplicative Inverse Property of Inequality refers to the principle that when you take the reciprocal of both sides of an inequality, you must reverse the direction of the inequality if both sides are positive. For example, if you have an inequality a > b (where both a and b are positive), taking the reciprocals becomes 1/a < 1/b.
This is because multiplying both sides of an inequality by a negative number or a reciprocal (which may imply a negative inversion) necessitates flipping the inequality. This property ensures that the relationship between the two quantities remains consistent after the operation. So, in context to the options provided by the student:
A) Taking the reciprocal of both a and b can change the direction of the inequality (Correct)
It's crucial to note that this property generally does not apply when dealing with square roots, absolute values, or logarithms in the same way, explaining why the other options provided are not applicable to the Multiplicative Inverse Property of Inequality. Those operations have their own specific impacts on inequalities.
