Inequalities Interval Notation
posted by Sasha
Solve: (2 + x)(5 + x)(4  x) < 0 (write answer in interval notation)
Which I think is xe (infinity, 5) or (2, 4)
and
Solve: (4x + 1)(3  x) > or equal to 0 (write answer in interval notation)
Which im unsure

Jai
#1.
(2 + x)(5 + x)(4  x) < 0
Equate each term inside the parenthesis to zero:
2 + x < 0
x < 2
5 + x < 0
x < 5
4  x < 0
4 < x
x > 4
When x > 4, only the term (4  x) is negative while the others are positive, so it satisfies the inequality.
When x < 5, say x = 6,
(2  6)(5  6)(4 + 6) < 0
4 * 1 * 10 < 0
40 < 0
which is false. So x cannot be less than or equal to 5.
When x < 2 but greater than 5, say x = 3
(2  3)(5  3)(4 + 3) < 0
1 * 2 * 7 < 0
14 < 0
which is true. So 5 < x < 2. Combine it with x > 4, we'll have
(5, 2) U (4, +infinity)
#2.
(4x + 1)(3  x) ≥ 0
Same procedure we'll do. Equate each term to zero:
4x + 1 ≥ 0
4x ≥ 1
x ≥ 1/4
3  x ≥ 0
3 ≥ x
We get the intersection. So x must be greater than or equal to 1/4 but less than or equal to 3:
[1/4 , 3]
hope this helps~ `u` 
Steve
After you get a bit of practice with this kind of stuff, you can feel confident using what you know about the graphs of polynomials.
(2 + x)(5 + x)(4  x) < 0
You know this is a cubic, with three roots. The graph crosses the xaxis at each root. So, pick a very large negative value for x.
The product ()()(+) is positive
So, you know that the graph crosses below the xaxis at 5 back up at 2 and back down at 4. So, f(x) < 0 on
(5,2)U(4,∞)
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