Solve the compound inequality. Graph the solutions. 2 < 5p - 3 <= 22 The solutions are given by (Type an inequality or a compound inequality. Simplify your answer.)
2 < 5p - 3 <= 22
Add 3 to all parts of the inequality:
2 + 3 < 5p - 3 + 3 <= 22 + 3
5 < 5p <= 25
Divide all parts of the inequality by 5:
5/5 < 5p/5 <= 25/5
1 < p <= 5
Therefore, the solution is 1 < p <= 5.
To solve the compound inequality 2 < 5p - 3 <= 22, we need to break it down into two separate inequalities and solve each one individually.
First, let's solve the left inequality: 2 < 5p - 3.
Adding 3 to both sides, we get: 5p - 3 + 3 > 2 + 3.
Simplifying, we have: 5p > 5.
Now, divide both sides by 5: (5p)/5 > 5/5.
The result is: p > 1.
Next, we solve the right inequality: 5p - 3 <= 22.
Adding 3 to both sides: 5p - 3 + 3 <= 22 + 3.
Simplifying, we have: 5p <= 25.
Dividing by 5: (5p)/5 <= 25/5.
The result is: p <= 5.
Now, we have the two inequalities: p > 1 and p <= 5.
To graph the solution, we can represent this on a number line. The solution is the overlapping region between p > 1 and p <= 5.
On the number line, put an open circle at 1 to represent p > 1 (since it does not include 1), and a closed circle at 5 to represent p <= 5 (since it includes 5). Then draw a shaded region between 1 and 5.
The graph should show a shaded region on the number line between 1 and 5, including 5 but not including 1.
Therefore, the solution is: 1 < p <= 5.
To solve the compound inequality 2 < 5p - 3 <= 22, we need to isolate the variable p.
First, let's solve the first inequality in the compound inequality: 2 < 5p - 3.
Adding 3 to both sides of the inequality, we get: 2 + 3 < 5p.
Simplifying this, we have: 5 < 5p.
Now, let's solve the second inequality: 5p - 3 <= 22.
Adding 3 to both sides of the inequality, we get: 5p <= 25.
Next, dividing both sides of the inequality by 5, we obtain: p <= 5.
So, we have two inequalities: 5 < 5p and p <= 5.
Combining these two inequalities, we can write the compound inequality as: 5 < 5p and p <= 5.
To graph the solutions, we will represent each inequality on a number line.
For the inequality 5 < 5p, we need to divide both sides of the inequality by 5 to get p alone. That gives us p > 1.
On the number line, we put an open circle at 1 (since 1 is not included in the solution), and draw an arrow pointing to the right to represent all the values of p greater than 1.
For the inequality p <= 5, we represent this by drawing a closed circle at 5 (since 5 is included in the solution), and draw an arrow pointing to the left to represent all the values of p less than or equal to 5.
Combining both inequalities on the number line, we shade the region between the two arrows representing the overlapping solutions.
In conclusion, the solutions to the compound inequality 2 < 5p - 3 <= 22 are represented graphically by the shaded region between the open circle at 1 and the closed circle at 5 on a number line.