algebra 2 help?

Question

6: Write in interval notation: x < 5 AND x < 3
(-∞ , 3)
(-∞ , 5)
(3, 5)
(-∞ , ∞ )


7: Solve and write in interval notation: -2 ≤ x + 4 OR -1 + 3x > -8
(-∞ , ∞ )
[-6, -3)
[-6, -7/3)
[-6, ∞ )


8: Solve and write in interval notation: x – 3 ≤ 2 AND -x / 3 < 2
(- ∞ , -2/3)
(-∞, ∞)
(-6, 5]
(-6, ∞)


9: Solve and write in interval notation: -3x > 2 OR (2x + 2) / 3 > 0
(-2/3, 1)
(-1, -2/3)
(-∞, -2/3)
(-∞, ∞)


10: Solve and write in interval notation: -3 ≤ (x – 4) / 2 < 4
(-∞, ∞)
[-2, 12)
[1, 8)
(-2, Inf)


11: Solve and write in interval notation: -7 < 4x – 3 ≤ 7
(-1, 5/2]
(-1, 1]
(-∞, ∞)
(-1, ∞)


12: Solve and write in interval notation: -4x – 2 < 6 AND 3x ≥ -6
[-2, ∞)
(-∞, ∞)
(-2, ∞)
(-∞, -2]

Answer 1

6. x < 5 AND x < 3 means that x has to satisfy both inequalities at the same time. The only way of doing that is x < 3, or (-∞ , 3).

7. -2 ≤ x + 4
-6 ≤ x

-1 + 3x > -8
3x > -7
x > -7/3

Since it's an or situation, x ≥ -6 answers both inequalities, so [-6, ∞ ).


8. x – 3 ≤ 2
x ≤ 5

-x / 3 < 2
-x < 6
x > -6

Since this is an and situation, find the area that these overlap: (-6, 5].

9. -3x > 2
x < -2/3

(2x + 2) / 3 > 0
2x + 2 > 0
2x > -2
x > -1

Since these overlap everywhere, and since this is an or situation, (-∞, ∞).

10. -3 ≤ (x – 4) / 2 < 4
-6 ≤ x - 4 < 8
-2 ≤ x < 12

Answer: [-2, 12).

11. -7 < 4x – 3 ≤ 7
-4 < 4x ≤ 10
-1 < x ≤ 5/2

Answer: (-1, 5/2].

12. -4x – 2 < 6
-4x < 8
x > -2

3x ≥ -6
x ≥ -2

Answer: (-2, ∞).