Solve the compound inequality. Graph the solutions. -5 < 2p - 3 <= 11
To solve the compound inequality -5 < 2p - 3 ≤ 11, we will solve the two separate inequalities and then combine the solutions.
First, let's solve 2p - 3 ≤ 11:
Add 3 to both sides:
2p - 3 + 3 ≤ 11 + 3
2p ≤ 14
Divide both sides by 2:
2p/2 ≤ 14/2
p ≤ 7
Next, let's solve -5 < 2p - 3:
Add 3 to both sides:
-5 + 3 < 2p - 3 + 3
-2 < 2p
Divide both sides by 2:
-2/2 < 2p/2
-1 < p
Now, let's combine the solutions.
The solution to -5 < 2p - 3 ≤ 11 is p ∈ (-1, 7].
This means that p is greater than -1 and less than or equal to 7.
To graph this solution on a number line, draw a closed circle at 7 and an open circle at -1. Then, shade the region between the two circles to represent the solution.
To solve the compound inequality -5 < 2p - 3 <= 11, we need to break it down into two separate inequalities and solve them individually.
First, let's consider the left inequality, -5 < 2p - 3:
-5 < 2p - 3
To isolate the variable p, we add 3 to both sides:
-5 + 3 < 2p - 3 + 3
-2 < 2p
Next, divide both sides by 2:
-2/2 < 2p/2
-1 < p
So, the left inequality is p > -1.
Next, let's consider the right inequality, 2p - 3 <= 11:
2p - 3 <= 11
To isolate the variable p, we add 3 to both sides:
2p - 3 + 3 <= 11 + 3
2p <= 14
Next, divide both sides by 2:
2p/2 <= 14/2
p <= 7
So, the right inequality is p <= 7.
Thus, the solution to the compound inequality -5 < 2p - 3 <= 11 is -1 < p <= 7.
To graph the solutions, we can use a number line. We represent the number -1 with an open circle (since it is not included in the solution), and the number 7 with a closed circle (since it is included in the solution). Then, we shade the area between -1 and 7 to represent all the values that satisfy the inequality.
Here is a graph of the solution on a number line:
------o==========o------
-1 7
The shaded region represents the values of p that satisfy the compound inequality.
To solve the compound inequality -5 < 2p - 3 <= 11, we need to break it down into two separate inequalities.
First, let's solve the left inequality, -5 < 2p - 3:
Add 3 to both sides of the inequality:
-5 + 3 < 2p - 3 + 3
-2 < 2p
Now, divide both sides of the inequality by 2:
-2/2 < 2p/2
-1 < p
So, the left inequality is -1 < p.
Next, let's solve the right inequality, 2p - 3 <= 11:
Add 3 to both sides of the inequality:
2p - 3 + 3 <= 11 + 3
2p <= 14
Now, divide both sides of the inequality by 2:
2p/2 <= 14/2
p <= 7
So, the right inequality is p <= 7.
Now, we have two inequalities: -1 < p and p <= 7.
To represent these inequalities on a number line, we draw a line, and mark -1 with an open circle (since it's not included in the solution) and shade towards the right side of the number line. Then, we mark 7 with a closed circle (since it is included in the solution) and shade towards the left side of the number line.
The graph would look like this:
-----o=====================o----
-1 7
So, the solutions to the compound inequality -5 < 2p - 3 <= 11 are all the values of p that lie between -1 and 7 (inclusive on 7).
In interval notation, the solution would be [-1,7].