4a-2 (less than or equal to) a + 1 (less than or equal to) 3a + 4
I'm assuming I should put this on a line, like I did with some other problems on my worksheet, right? But I have no idea how I would show 4a-2 on a line! I have no idea how to do this.
Thank you so much!
I think I understand how to do this... here's my work.
4a - 2 < a + 1 < 3a + 4
I separated them into two pieces:
4a-2 < a + 1 and a + 1 < 3a + 4
Then, I added two to each side of the first one, and took away 1 from each side of the second one.
4a < a + 3 and a < 3a + 3
I took away took away a from each side of the first one and 3a from each side of the second one.
3a < 3 and -2a + 3
Then, I will put it on a line. Does my work look right?
4a-2<a+1
3a<3 or a<1
a+1<3a+4
-3<2a
-3/2 < a
To show the inequality 4a-2 ≤ a+1 ≤ 3a+4 on a line, you can represent the values of a on the number line. Here's how you can do it:
1. Start by drawing a horizontal line representing the number line.
2. Choose a point and label it with the value of a. This will be your starting point.
3. Determine the range for a. In this case, we have 4a-2 ≤ a+1 ≤ 3a+4. To simplify it, let's split it into two separate inequalities:
4a-2 ≤ a+1 (First inequality)
a+1 ≤ 3a+4 (Second inequality)
4. Solve the first inequality: 4a-2 ≤ a+1:
Subtract 'a' from both sides:
4a-2-a ≤ a+1-a
3a-2 ≤ 1
Add '2' to both sides:
3a ≤ 3
Divide both sides by '3':
a ≤ 1
This means that 'a' can take values less than or equal to 1.
5. Represent the first inequality on the number line:
Place a closed dot at the point you initially labeled for 'a' to represent the equality a = 1. Then, draw an arrow to the left to show that 'a' can take any value less than or equal to 1.
6. Solve the second inequality: a+1 ≤ 3a+4
Subtract 'a' from both sides:
a+1-a ≤ 3a+4-a
1 ≤ 2a+4
Subtract '4' from both sides:
1-4 ≤ 2a+4-4
-3 ≤ 2a
Divide both sides by '2':
-3/2 ≤ a
This means that 'a' can take values greater than or equal to -3/2.
7. Represent the second inequality on the number line:
Place a closed dot at the point you initially labeled for 'a' to represent the equality a = -3/2. Then, draw an arrow to the right to show that 'a' can take any value greater than or equal to -3/2.
8. Finally, combine the two representations on the number line by including all the values between -3/2 and 1. This will be the shaded region.
By following these steps, you will have accurately represented the inequality 4a-2 ≤ a+1 ≤ 3a+4 on a number line.