Solve the compound inequality. Graph the solutions.
1 < 2p - 3 < 13
To solve the compound inequality, we will solve the individual inequalities and then combine the solutions.
1 < 2p - 3:
Adding 3 to both sides, we get:
1 + 3 < 2p - 3 + 3
4 < 2p
Dividing both sides by 2, we get:
4/2 < 2p/2
2 < p
2p - 3 < 13:
Adding 3 to both sides, we get:
2p - 3 + 3 < 13 + 3
2p < 16
Dividing both sides by 2, we get:
2p/2 < 16/2
p < 8
The solution for the compound inequality is 2 < p < 8.
To graph the solutions on a number line, we will plot a closed circle at 2 and an open circle at 8, and shade the region between them.
The graph is as follows:
```
-∞ ┌─────┲━━━━━o─────o━━━━━┱─────┐ ∞
2 8
```
To solve the compound inequality 1 < 2p - 3 < 13, we will break it down into two separate inequalities and solve each one individually.
1 < 2p - 3 and 2p - 3 < 13
Let's solve the first inequality:
1 < 2p - 3
Adding 3 to both sides, we get:
4 < 2p
Dividing both sides by 2, we get:
2 < p
Now, let's solve the second inequality:
2p - 3 < 13
Adding 3 to both sides, we get:
2p < 16
Dividing both sides by 2, we get:
p < 8
Therefore, the compound inequality can be written as:
2 < p < 8
To graph the solutions, we first draw a number line. Then, we mark the points 2 and 8 on the number line. Since the inequality is strict, we use open circles at these points. Finally, we shade the region between the two open circles.
The graphical representation of the solutions to 1 < 2p - 3 < 13 is:
o o
|---------|---------|
-1 2 8 13
To solve the compound inequality 1 < 2p - 3 < 13, we need to break it down into two separate inequalities and solve each one individually.
The given compound inequality has two parts connected by the "and" operator. Let's solve each part separately:
1. Solve the first inequality: 1 < 2p - 3
Add 3 to both sides to isolate the variable:
1 + 3 < 2p - 3 + 3
4 < 2p
Now, divide both sides by 2 to solve for p:
4/2 < 2p/2
2 < p
So, the solution to the first inequality is p > 2.
2. Solve the second inequality: 2p - 3 < 13
Add 3 to both sides to isolate the variable:
2p - 3 + 3 < 13 + 3
2p < 16
Now, divide both sides by 2 to solve for p:
2p/2 < 16/2
p < 8
Therefore, the solution to the second inequality is p < 8.
Now, we can combine the two solutions to get the overall solution for the compound inequality:
p > 2 and p < 8
To graph the solutions, we need to draw a number line and shade the region between 2 and 8, including the endpoints. Since the inequality is strict ( ">" and "<"), we should use open circles at 2 and 8 to indicate that they are not part of the solution.
On the number line, mark a point at 2 and 8. Then draw an open circle at 2 and 8. Finally, shade the region between these two points.
The graph should show an open circle at 2 and 8, and a shaded region between them on the number line. This represents the solution to the compound inequality 1 < 2p - 3 < 13.