1)
v - 6 ≥ 4
+6 +6
v ≥ 10
Graph: 10 -----------------→ the dot at 10 is filled in because of the "equal to" symbol
********************************************************************************************************
2)
-5x < 15
[tex]\frac{-5x}{-5} > \frac{15}{-5}[/tex] divided by a negative so the symbol flipped
x > -3
Graph: -3 o------------------→ the dot at -3 is NOT filled in because it's NOT "equal to"
********************************************************************************************************
3)
3k > 5k + 12
-5k -5k
-2k > 12
[tex]\frac{-2k}{-2} < \frac{12}{-2}[/tex] divided by a negative so the symbol flipped
k < -6
Graph: ←------------o -6 the dot at -6 is NOT filled in because it's NOT "equal to"
********************************************************************************************************
5)
2t ≤ 4 or 7t ≥ 49
[tex]\frac{2t}{2} \leq\frac{4}{2}[/tex] or [tex]\frac{7t}{7} \geq \frac{49}{7}[/tex]
t ≤ 2 or t ≥ 7
Graph: ←------- 2 7 ---------→ the dots at 2 and 7 are filled in
********************************************************************************************************
6)
| n + 2 | = 4
n + 2 = 4 or n + 2 = -4
-2 -2 -2 -2
n = 2 or n = -6
Answer: n = {2, -6}
********************************************************************************************************
7)
| 2x - 7 | > 1
2x - 7 > 1 or 2x - 7 < -1
+7 +7 +7 +7
2x > 8 or 2x < 6
[tex]\frac{2x}{2} > \frac{8}{2}[/tex] or [tex]\frac{2x}{2} < \frac{6}{2}[/tex]
x > 4 or x < 3
Graph: ←----------o 3 4 o------------→
********************************************************************************************************
8)
A = {1, 2, 3, 4, 5, 6, 7, 8, 9} B = {2, 4, 6, 8}
A U B = {1, 2, 3, 4, 5, 6, 7, 8, 9} union combines both sets
A ∩ B = {2, 4, 6, 8} intersection includes only those that are in BOTH sets
********************************************************************************************************
9)
P = {1, 5, 7, 9, 13} R = { 1, 2, 3, 4, 5, 6, 7} Q = {1, 3, 5}
P ∩ R ∩ Q = {1, 5}
P ∩ R = {7} disregard 1 & 5 since they are already in P∩Q∩R
R ∩ Q = {3} disregard 1 & 5 since they are already in P∩Q∩R
P ∩ Q = { } disregard 1 & 5 since they are already in P∩Q∩R
P no intersection = {9, 13)
R no intersection = {2, 4, 6}
Q no intersection = { }
If you can't figure out how to draw it based on the information I provided, please see the attached diagram. Note: Usially, you would only identify P, Q, R. I identified the intersections so you would understand why those specific numbers are placed in the designated sections.
