Problem Solving

Thanks for your help in advance!

{A} x^2 >=1
If we consider just the positive values
x >=1
So, there is no end limit to line
NO

{B} x^3 <=27
If we consider just the positive values
x <=3
So, there is no end limit to line
NO

{C} x^2>=16
If we consider just the positive values
x >=4
So, there is no end limit to line
NO

{D} 2 <= |X| <= 5
lets break:
|x| >= 2
==> x > =2 or x <= -2
No finite length
|x| < = 5
==> x <= 5 or x >= -5
NO

{E}
2 <= 3x + 4 <= 6
subtract 4 from all sides
-2 <= 3x <= 2
Divide by 3
-2/3 <= x <= 2/3
So, the line segment has finite length
YES

Answer [spoiler]{E}[/spoiler]

Which of the following inequalities has a solution set that, when graphed on the number line is a single line segment of finite length?

1. x�> 1
2. x³ ≤ 27
3. x² ≥ 16
4. 2 ≤ |x| ≤ 5
5. 2 ≤ 3x+4 ≤ 6

When the question stem requires that all 5 answer choices be considered, the correct answer choice is likely to be D or E (so that the average test-taker will waste time checking all 5 answers).
Let's start with E and work our way up.

Answer choice E: 2 ≤ 3x+4 ≤ 6
Subtract 4 from each part:
-2 ≤ 3x ≤ 2
Divide each part by 3:
-2/3 ≤ x ≤ 2/3.

-2/3 ≤ x ≤ 2/3 includes every value between -2/3 and 2/3, inclusive.
This range has two ENDPOINTS: -2/3 and 2/3.
Thus, when graphed on the number line, the solution set will yield a line segment with two endpoints -- in other words, a line segment of FINITE LENGTH:
-2/3--------------------2/3

The correct answer is E.