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Absolutely Value Inequality

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Absolutely Value Inequality

by Buix0065 » Sun May 08, 2011 6:08 pm
Hello,

I came across the following problem:

A number line with endpoints at -5 --------3

Which of the following inequalities is an algebraic expression for the shaded part of the number line above? (again, the shaded part is -5 through 3)

A. |x| ≤ 3
B. |x| ≤ 5
C. |x - 2| ≤ 3
D. |x - 1| ≤ 4
E. |x + 1| ≤ 4

I don't quite understand what this problem is getting at. If you're just plugging in numbers to find the equation that is true, then both B and E would be true. The book talks about the mid-point, I don't see how this comes into play.

Any insights would be appreciated, thanks!

OA: E
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by vineeshp » Sun May 08, 2011 6:43 pm
B cannot be the answer.For values 4,-4 the equation goes out of play.
-5≤x≤3
A. |x| ≤ 3
Equation: -3≤x≤3
B. |x| ≤ 5
-5≤x≤5
C. |x - 2| ≤ 3
When x-2≤3 -> x≤5
x-2≤0
-(x-2)≤3 -> -1 ≤ x
Combined eqn not the answer.


D. |x - 1| ≤ 4
x-1≤5 -> x≤5 - Wrong.

E. |x + 1| ≤ 4
This has to be right. All prev were wrong.
x + 1 ≤ 4
implies x ≤ 3
which forms the right side of the inequality.
-(x+1) ≤ 4
-x-1 ≤ 4
-x ≤ 5
-5 ≤ x
which forms the other side of the inequality
Vineesh,
Just telling you what I know and think. I am not the expert. :)
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by Stuart@KaplanGMAT » Sun May 08, 2011 11:12 pm
Buix0065 wrote:Hello,

I came across the following problem:

A number line with endpoints at -5 --------3

Which of the following inequalities is an algebraic expression for the shaded part of the number line above? (again, the shaded part is -5 through 3)

A. |x| ≤ 3
B. |x| ≤ 5
C. |x - 2| ≤ 3
D. |x - 1| ≤ 4
E. |x + 1| ≤ 4
Hi!

I've bolded the key words in the question to emphasize what we're really being asked.

While you're correct that the inequality in (B) includes all of the points in the shaded part of the number line, it's also true that (B) includes some points that aren't in the shaded region, e.g. +4 and +5. Since we've been asked for an algebraic equation for the shaded region, we need to choose the answer that's a precise match.

So, the correct answer must include every point inside the shaded region but not include any points not inside the shaded region.

Vineeshp did a great job illustrating both how we could solve by either picking numbers or algebra, so I won't touch on either of those unless you have specific questions about those approaches.
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by kevincanspain » Sun May 08, 2011 11:36 pm
It is useful to remember that |x - y| represents the distance between x and y.

The line segment from - 5 to 3 is 8 units long and its midpoint is -1.

It is the set of all points that are at most 4 units from - 1, and thus can be so written:

| x -(-1)| <= 4
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