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# Inequalities

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Intern
Joined: 08 Jun 2009
Posts: 33

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16 Jun 2009, 07:20
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Which of the following inequalities has a solution set that, when graphed on the number line, is a single line segment of finite length?

$$(A) x^4 >= 1$$
$$(B) x^3 <= 27$$
$$(C) x^2 >= 16$$
$$(D) 2 <= |x| <= 5$$
$$(E) 2 <= 3x + 4 <= 6$$
Current Student
Joined: 03 Aug 2006
Posts: 115

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16 Jun 2009, 08:11
If you scan through the answers it seems that E is the answer but lets try them out.

We can break down each one to see what x equals...

$$\text A. x^4 \geq 1$$

$$\Rightarrow x \geq 1 or x \leq -1$$

This is not a finite segment. Not the answer.

$$\text B. x^3 \leq 27$$

$$\Rightarrow x \leq 3$$

This is not a finite segment. Not the answer.

$$\text C. x^2 \geq 16$$

$$\Rightarrow x \geq 4 \text{ or } x \leq -4$$

This is not a finite segment. Not the answer.

$$\text D. 2 \leq |x| \leq 5$$

$$\Rightarrow 2 \leq x \leq 5 \text{ or } -2 \geq x \geq -5$$

Again this is not a finite segment. Not the answer.

$$\text E. 2 \leq 3x+4 \leq 6$$

$$\Rightarrow -2 \leq 3x \leq 2$$

$$\Rightarrow -2/3 \leq x \leq 2/3$$

This is a finite line segment. Hence the answer is E.
Intern
Joined: 08 Jun 2009
Posts: 33

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16 Jun 2009, 17:17
OA is E.

I need some help in understanding why D is not the answer. From my interpretation, |x| tells us it will be positive regardless of the sign, meaning the inequality is a finite segment from 2 to 5.
Current Student
Joined: 03 Aug 2006
Posts: 115

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16 Jun 2009, 19:17
When there is an absolute value you need to split the equation into positive and negative cases. Because of that we have two ranges for x for option D.

Here is a good overview of absolute values.
http://www.purplemath.com/modules/solveabs.htm
Re: Inequalities   [#permalink] 16 Jun 2009, 19:17
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