Mathematics, 07.10.2020 17:01 lyndamahe0

Determine whether the absolute value of a difference is sometimes, always, or never the same as the difference of the absolute values. Ia-bI = IaI - IbI

Answers: 3

Show answers

Answers

Answer from: quinten3700

|a-b| = |a| - |b| if and only if $a \ge 0\,\land\,b\ge 0$ . That is, it is sometimes the same.

Absolute Value is a mathematic Form which represents either the Magnitude or the Distance respect to Origin of a given Real Number, whose is depicted below:

$|x| = \left \{ {{x,\,x\ge 0} \atop {-x,\,x< 0}} \right.$ , for all $x\in \mathbb{R}$ (1)

In this question we must check if |a|-|b| = |a - b| . Hence, we must check it for the following four cases:

(i) $a \ge 0\,\land\,b\ge 0$

(ii) $a\ge 0\,\land \,b < 0$

(iii) $a < 0\,\land\,b$

(iv) $a< 0\,\land b\,\ge 0$

Case I

1) $|a| \ge 0$ , $|b| \ge 0$ , $|a-b| \ge 0$ Definition of absolute value.

2) $a \ge 0\,\land\,b\ge 0$ Given.

3) a - b = a - b Definition of absolute value/Reflexive Property

4) |a-b| = |a| - |b| Result.

Case II

1) $|a| \ge 0$ , $|b| \ge 0$ , $|a-b| \ge 0$ Definition of absolute value.

2) $a\ge 0\,\land \,b < 0$ Given.

3) $a - b =^{?} a - (-b)$ Definition of absolute value.

4) $a - b =^{?} a + b$ -(-x) = x

5) $-b =^{?} b$ Compatibility with addition/Modulative property/Existence of additive inverse/Absurd found.

6) $|a-b| \ne |a| - |b|$ Reductio ad absurdum.

Case III

1) $|a| \ge 0$ , $|b| \ge 0$ , $|a-b| \ge 0$ Definition of absolute value.

2) $a < 0\,\land\,b$ Given.

3) $a-b =^{?} -a-(-b)$ Definition of absolute value.

4) $a-b =^{?} -(a -b)$ $(-1)\cdot x = -x$ /Distributive property/Absurd found.

5) $|a-b| \ne |a| - |b|$ Reductio ad absurdum.

Case IV

1) $|a| \ge 0$ , $|b| \ge 0$ , $|a-b| \ge 0$ Definition of absolute value.

2) $a< 0\,\land b\,\ge 0$ GIven.

3) $a-b =^{?} -a - b$ Definition of absolute value.

4) $a =^{?} -a$ Compatibility with addition/Modulative property/Existence of additive inverse/Absurd found.

5) $|a-b| \ne |a| - |b|$ Reductio ad absurdum.

In a nutshell, |a-b| = |a| - |b| if and only if $a \ge 0\,\land\,b\ge 0$ . That is, it is sometimes the same.

Please see this question related to Absolute Values: link

Answer from: goly9453

Sometimes True

Explanation:

Let's see what happens when a = b

Let's pick something like a = 2, so b = 2.

|a-b| = |2-2| = |0| = 0

|a|-|b| = |2|-|2| = 2-2 = 0

Both result in 0.

We see that |a-b| = |a|-|b| is true for this example.

The given equation is either "sometimes true" or "always true"

Now let's pick something where a < b. Let a = 1 and b = 2

|a-b| = |1-2| = |-1| = 1

|a|-|b| = |1|-|2| = 1-2 = -1

We get different results, so |a-b| = |a|-|b| is not true for this example.

So this leads us to say the original equation is sometimes true.

Answer from: Quest

homozygous. one is homozygous, while two is heterozygous.

Answer from: Quest

answer: my brother can figure it out, imma go get him so i can answer : ) edit: my brother is sleeping, sorry man

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 12:50

Solve the system by the elimination method. 2x + y - 4 = 0 2x - y - 4 = 0 when you eliminate y, what is the resulting equation? a.x = 0 b.4x = -8 c.4x = 8

Answers: 1

Answer

Mathematics, 21.06.2019 16:30

Jorge planted flowers in his garden he planted one row of 12 tulips and one row of 36 daisies george's friend kylie has a garden with 75 flowers in it kylie's garden only contains tulips and daisies. is it possible for kylie‘s garden she have the same ratio of tulips to daisies as george's garden?

Answers: 1

Answer

Mathematics, 21.06.2019 17:30

Which equation represents a line that is parallel to the line whose equation is 3x-2y=7( show all work)

Answers: 3

Answer

Mathematics, 21.06.2019 23:50

Which statement explains how you could use coordinate geometry to prove the opposite sides of a quadrilateral are congruent? a. use the slope formula to prove the slopes of the opposite sides are the same. b. use the slope formula to prove the slopes of the opposite sides are opposite reciprocals. c. use the distance formula to prove the lengths of the opposite sides are the same. d. use the distance formula to prove the midpoints of the opposite sides are the same.

Answers: 3

Answer

You know the right answer?

Determine whether the absolute value of a difference is sometimes, always, or never the same as the...

Questions

Mathematics, 18.02.2021 19:20

Write an expression in the simplest form that represents the perimeter of the following shape

Mathematics, 18.02.2021 19:20

Juan visited a candy store and purchased 7 pounds of candy consisting of Atomic Fireballs and Gummi Bears. Atomic Fireballs cost $4.25 per pound and G...