Answer:
3. a ≤ 7, a > 3; <-------------7 - - - - - - - 3------------->
4. d ≤ 10, d < 9; <-----------------------10---9- - - - - - - - - ->
5. 8t ≤ 360, t ≤45
Explanation:
3. 6a ≤ 42 < divide both sides by 6 to get a by itself
a ≤ 7 < this is correct
a + 4 > 7 < subtract 4 from both sides to isolate a
a > 3 this is also correct!
now to graph these inequalities on the number line, we will do them one at a time
a ≤ 7, you would put a closed circle (or a bracket) on 7, as the answer is included in the solution, and you would shade to the left of 7, as a has to be less than or equal to 7 in order for the inequality to be true
< ----------7 - - - - >
now for a > 3,you would shade numbers to the right of 3, as a has to be greater than 3 in order for the inequality to be true. you would put an open circle (or a parentheses) on the number 3, as its not included in the solution as shown by the inequality symbol
< - - - - 3--------->
together on a number line, both inequalities would look like the following:
<-------------7 - - - - - - - 3------------->
the middle section is not included in either of the inequalities
4. d - 8 ≤ 2 < add 8 to both sides
d ≤ 10 < in your original answer, you forgot to add 8 to the right side of the equation, remember that whatever you do to the left you also do to the right side
on a number line, we would shade to the left of 10, as any answer for d would have to be less than or equal to 10 in order for the inequality to be true. again, a closed circle (or bracket) would be on the number 10 as it is included in the solution
< ----------10 - - - - >
9d < 81 <-- divide both sides by 9 to get d alone
d < 9 < this is correct in your original answer
on a number line, we would shade to the left of 9, as any number inserted into the inequality for d has to be less than 9 in order for the inequality to be true. we would put an open circle (or parentheses) on the number 9 because it is not included in the solution
< ----------9 - - - - >
together on a number line, both terms would look like this:
<-----------------------10---9- - - - - - - - - ->
since both are being shaded to the left, both would have a line going to the left
5. banquet room holds 360 people, with 8 people per table (t)
8t ≤ 360 < this is the correct inequality, as we cannot exceed 360 people in the room, and we can seat 8 people at each table
now we divide both sides by 8 in the inequality to get t alone
t ≤ 45
so there can only be 45 tables or less in the entire banquet in order to not exceed the 360 person capacity
on a number line the inequality looks like the following:
< --------45 - - - >
45 is included in the solution as we can have 45 tables for 8 people each, and we would be at max capacity in the banquet. 45 on the number line would have a closed circle (or bracket) on the number itself
hope this helped!