Express Without Absolute Value

[harpazo]

Rewrite each expression in a form that does not contain absolute values.
How is this done?


(a) | pi - 4 | + 1


(b) | x - 5 | given that x > or = 5


(c) | t - 5 | given that t < 5.

 

[JackKingstone]

Hi,

To rewrite each expression in a form that does not contain absolute values, we need to interpret what the absolute value represents geometrically and then apply the given conditions.

(a) ∣π−4∣+1| \pi - 4 | + 1∣π−4∣+1:

The absolute value ∣π−4∣| \pi - 4 |∣π−4∣ represents the distance between π\piπ and 444 on the number line. Since π\piπ is approximately 3.141593.141593.14159, we have:

∣π−4∣=∣3.14159−4∣=∣−0.85841∣=0.85841| \pi - 4 | = | 3.14159 - 4 | = | -0.85841 | = 0.85841∣π−4∣=∣3.14159−4∣=∣−0.85841∣=0.85841

Therefore,

∣π−4∣+1=0.85841+1=1.85841| \pi - 4 | + 1 = 0.85841 + 1 = 1.85841∣π−4∣+1=0.85841+1=1.85841

So, ∣π−4∣+1| \pi - 4 | + 1∣π−4∣+1 simplifies to 1.858411.858411.85841.

(b) ∣x−5∣| x - 5 |∣x−5∣ given that x≥5x \geq 5x≥5:

Here, ∣x−5∣| x - 5 |∣x−5∣ represents the distance between xxx and 555 on the number line. Since x≥5x \geq 5x≥5, xxx is to the right of 555 or exactly at 555. Therefore,

∣x−5∣=x−5| x - 5 | = x - 5∣x−5∣=x−5

So, ∣x−5∣| x - 5 |∣x−5∣ simplifies to x−5x - 5x−5.

(c) ∣t−5∣| t - 5 |∣t−5∣ given that t<5t < 5t<5:

In this case, ∣t−5∣| t - 5 |∣t−5∣ represents the distance between ttt and 555 on the number line, but since t<5t < 5t<5, ttt is to the left of 555. Therefore,

∣t−5∣=−(t−5)=5−t| t - 5 | = -(t - 5) = 5 - t∣t−5∣=−(t−5)=5−t

So, ∣t−5∣| t - 5 |∣t−5∣ simplifies to 5−t5 - t5−t

I hope this will be helpful.

Jack