Solve the compound inequality. Graph the solutions. -5 < 2p - 3 <= 9
We can solve this compound inequality by working on each inequality separately and then taking the intersection of their solutions.
Starting with the left inequality -5 < 2p - 3:
Adding 3 to both sides gives us: -2 < 2p
Dividing both sides by 2 gives us: -1 < p
Now, let's solve the right inequality 2p - 3 <= 9:
Adding 3 to both sides gives us: 2p <= 12
Dividing both sides by 2 gives us: p <= 6
So, now we have the solution:
-1 < p <= 6
To graph this solution, we'll draw a number line with an open circle at -1 to represent the "<" inequality, and a closed circle at 6 to represent the "≤" inequality. Then we'll shade the portion of the number line between -1 and 6.
Here's a rough sketch of the graph:
-1 0 1 2 3 4 5 6
----o----|----|----|----|----|----o----
The shaded region should be between the two points "-1" and "6" on the number line.
To solve the compound inequality -5 < 2p - 3 <= 9, we will first solve each inequality separately and then combine the solutions.
1. Solve the inequality 2p - 3 > -5:
Add 3 to both sides to isolate the variable:
2p - 3 + 3 > -5 + 3
2p > -2
Divide both sides by 2 to solve for p:
2p/2 > -2/2
p > -1
2. Solve the inequality 2p - 3 <= 9:
Add 3 to both sides to isolate the variable:
2p - 3 + 3 <= 9 + 3
2p <= 12
Divide both sides by 2 to solve for p:
2p/2 <= 12/2
p <= 6
Now, we have the individual solutions:
-1 < p <= 6
To graph the solutions, we can represent the solution set on a number line. Draw a number line and mark -1 and 6 on it. Since the compound inequality includes less than or equal to, we will use closed circles at -1 and 6 to indicate that they are included in the solution set. Then, draw a straight line segment between them. Shade the portion between -1 and 6 to represent the solutions of the compound inequality.
-1 6
*----o======
The shaded area represents the solutions: -1 < p <= 6.
To solve the compound inequality -5 < 2p - 3 <= 9, we need to break it down into two separate inequalities and solve them individually.
First, let's solve the inequality -5 < 2p - 3:
Add 3 to both sides of the inequality:
-5 + 3 < 2p - 3 + 3
-2 < 2p
Next, divide both sides by 2 (since 2p is multiplied by 2):
-2/2 < 2p/2
-1 < p
Now, let's solve the inequality 2p - 3 <= 9:
Add 3 to both sides of the inequality:
2p - 3 + 3 <= 9 + 3
2p <= 12
Next, divide both sides by 2:
(2p)/2 <= 12/2
p <= 6
Therefore, the solutions for the compound inequality -5 < 2p - 3 <= 9 are -1 < p ≤ 6.
To graph the solutions, we represent the inequality on a number line. We draw an open circle on the number -1 to represent the < inequality, and a closed circle on the number 6 to represent the ≤ inequality. Then, we shade the region between -1 and 6 to represent all the values of p that satisfy the inequality.