( / means absolute value)
Solve: 3+/2y-1/>=1. Graph the solution set on a number line.
The number line I have is:
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
First solution:
3+2y-1>=1
2y-4>=1
2y>=5
y>=5/2
I'm stuck b/c I had to correct the first part. I had a whole different answer. The graph for this would be an open circle between 2 and 3 and the arrow going right.
|2y-1| = +(2y-1) if y >1/2
|2y-1| = -(2y-1) if y <1/2
so do the whole problem first for y >1/2
3 + 2y - 1 >/= 1
2 + 2y >/= 1
1+y >/= 1/2
y >/= -1/2
so this is true if y >1/2
now do the whole problem for y<1/2
3 - 2y +1 - 1 >/= 1
3 - 2y >/= 1
-2y >/= -2
y </= 1
so it is true for any y greater than 1/2 and for any y less than 1 and so the entire real number line
Try some spots
y = 0 yes
y = -1 yes
y = +1 yes
y = 1/2 yes
y = -1/2 yes etc etc etc
To solve the inequality 3 + |2y - 1| ≥ 1, we need to consider two cases:
Case 1: 2y - 1 ≥ 0
In this case, the absolute value is equivalent to its argument, so we can rewrite the inequality as 3 + 2y - 1 ≥ 1:
2y + 2 ≥ 1
2y ≥ -1
y ≥ -1/2
Case 2: 2y - 1 < 0
In this case, the absolute value becomes the negative of its argument, so we need to flip the inequality sign when we remove the absolute value. Rewrite the inequality as 3 - (2y - 1) ≥ 1:
3 - 2y + 1 ≥ 1
-2y + 4 ≥ 1
-2y ≥ -3
y ≤ 3/2
Now, let's consider the graph on the number line.
In Case 1 (2y - 1 ≥ 0), we found that y ≥ -1/2. This means that all values of y greater than or equal to -1/2 satisfy the inequality. On the number line, we mark -1/2 with a closed circle (since it's included in the solution), and draw an arrow to the right to represent any value greater than -1/2.
In Case 2 (2y - 1 < 0), we found that y ≤ 3/2. This means that all values of y less than or equal to 3/2 satisfy the inequality. On the number line, we mark 3/2 with a closed circle (since it's included in the solution), and draw an arrow to the left to represent any value less than 3/2.
Combining both cases, we have the solution set represented on the number line as follows:
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
┃------●==============➝
-1/2 3/2
The closed circle represents -1/2, and the arrow to the right represents values greater than -1/2. The closed circle represents 3/2, and the arrow to the left represents values less than 3/2.