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# If | x + 1 | <= 5 and | y — 1| <= 5, what is the least

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If | x + 1 | <= 5 and | y — 1| <= 5, what is the least [#permalink]  05 Mar 2017, 13:03
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Question Stats:

63% (01:00) correct 36% (02:03) wrong based on 19 sessions

If $$| x + 1 | \leq 5$$ and $$|y — 1| \leq 5$$, what is the least possible value of the product $$xy$$ ?

[Reveal] Spoiler:
$$-36$$

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Re: If | x + 1 | <= 5 and | y — 1| <= 5, what is the least [#permalink]  11 Mar 2017, 21:07
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Explanation

Find $$| x + 1 | \leq 5$$:

$$-5 \leq x + 1 \leq 5$$ or $$-6 \leq x \leq 4$$

Now calculate $$| y - 1 | \leq 5$$:

$$-5 \leq y - 1 \leq 5$$ or $$-4 \leq y \leq 6$$

To find the least possible value of the product xy, put numbers from the extremes of x and y. Keep in mind that two positive numbers and two negative numbers will result in a positive number.

Since we want the least value, we are looking for a negative number and must use one positive and one negative value to obtain it.

If x = 4 and y = –4, then xy = –16
If x = –6 and y = 6, then xy = –36

The least possible value is –36.
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Sandy
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Re: If | x + 1 | <= 5 and | y — 1| <= 5, what is the least [#permalink]  08 Mar 2019, 11:13
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Carcass wrote:

If $$| x + 1 | \leq 5$$ and $$|y — 1| \leq 5$$, what is the least possible value of the product $$xy$$ ?

[Reveal] Spoiler:
$$-36$$

When solving inequalities involving ABSOLUTE VALUE, there are 2 things you need to know:
Rule #1: If |something| < k, then –k < something < k
Rule #2: If |something| > k, then EITHER something > k OR something < -k
Note: these rules assume that k is positive

GIVEN: |x + 1| ≤ 5
From Rule #1 above, we can write: -5 ≤ x + 1 ≤ 5
Subtract 1 from all sides to get: -6 ≤ x ≤ 4
So, x can be ANY value from -6 to 4 INCLUSIVE

GIVEN: |y - 1| ≤ 5
From Rule #1 above, we can write: -5 ≤ y - 1 ≤ 5
Add 1 to all sides to get: -4 ≤ y ≤ 6
So, y can be ANY value from -4 to 6 INCLUSIVE

What is the least possible value of the product xy?
The LEAST product will be NEGATIVE
So, one value (x or y) must be POSITIVE, and the other value must be NEGATIVE
The LEAST product occurs when x = -6 and y = 6
So, the least product = (-6)(6) = -36

Cheers,
Brent
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Re: If | x + 1 | <= 5 and | y — 1| <= 5, what is the least   [#permalink] 08 Mar 2019, 11:13
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