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

yep

step-by-step explanation:

Answer from: Quest

answer:

c. the third graph

Another question on Mathematics

Mathematics, 20.06.2019 18:02

What is the measure of x? 43° 86° 172° 344°

Answers: 1

Answer

Mathematics, 21.06.2019 16:00

Which is the graph of the piecewise function f(x)? f(x)=[tex]f(x) = \left \{ {{-x+1, x\leq 0 } \atop {x+1, x\ \textgreater \ 0}} \right.[/tex]

Answers: 3

Answer

Mathematics, 21.06.2019 18:30

Acone with volume 1350 m³ is dilated by a scale factor of 1/3. what is the volume of the resulting cone? enter your answer in the box. m³

Answers: 1

Answer

Mathematics, 21.06.2019 19:00

Rob spent 25%, percent more time on his research project than he had planned. he spent an extra h hours on the project. which of the following expressions could represent the number of hours rob actually spent on the project? two answers

Answers: 1

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, 04.06.2020 21:57

Finds the missing values of the variables. the diagram is not to scale.