Solve the double inequality
0<(x+3)/2<5
0 < (x+3)/2 < 5
multiply by 2
0 < x + 3 < 10
subtract 3
-3 < x < 7
To solve the double inequality 0 < (x + 3)/2 < 5, we will break it down into two separate inequalities and solve each one individually.
First, let's consider the left inequality: 0 < (x + 3)/2.
To solve it, we will multiply both sides of the inequality by 2, keeping in mind that when multiplying or dividing an inequality by a negative number, we need to reverse the direction of the inequality.
0 < (x + 3)/2
0 * 2 < ((x + 3)/2) * 2
0 < (x + 3)
Next, let's consider the right inequality: (x + 3)/2 < 5.
To solve it, we will multiply both sides of the inequality by 2, again keeping in mind the direction of the inequality.
(x + 3)/2 < 5
((x + 3)/2) * 2 < 5 * 2
(x + 3) < 10
So we have two separate inequalities:
0 < (x + 3)
(x + 3) < 10
Now let's solve each inequality individually:
1. Solving 0 < (x + 3):
Subtract 3 from both sides of the inequality:
0 - 3 < (x + 3) - 3
-3 < x
Therefore, x > -3.
2. Solving (x + 3) < 10:
Subtract 3 from both sides of the inequality:
(x + 3) - 3 < 10 - 3
x < 7
Therefore, x < 7.
Combining the individual solutions, we have:
-3 < x < 7
So the solution to the double inequality 0 < (x + 3)/2 < 5 is -3 < x < 7.