The Absolute Warm-up

Algebra Level 2

y = ∣ ∣ ∣ x − 5 ∣ + 5 ∣ − 5 ∣ + 5 y = \Big| \big| \small| x - 5 \small| + 5 \big| - 5 \Big| + 5

What is the smallest possible value of y y ?

Notation : ∣ ⋅ ∣ | \cdot | denotes the absolute value function .

-5 0 5 10

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Starbuck Beagley by Starbuck Beagley , 40, USA

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2 solutions

Shaun Leong
Jul 22, 2016

y y can be written as ∣ c ∣ + 5 |c| + 5 .

Since ∣ c ∣ ≥ 0 |c| \geq 0 , y = ∣ c ∣ + 5 ≥ 5 y=|c| + 5\geq 5 .

Indeed, this can be achieved when x = 5 x=5 .

17 Helpful 0 Interesting 3 Brilliant 0 Confused

Friend,please explain with some arguments with their supports for better understanding.

Hiren Patel - 4 years, 10 months ago

From the innermost |x_5|, no negative value can be obtained.
So least value is 0 when x+5.
Next |...| is |0+5|=5.
Last absolute value bounds are |5-5|=0.
So y=0+5=5.


1 Helpful 0 Interesting 0 Brilliant 0 Confused

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