solve an inequality and indicate the solution set, in interval notation. i need the steps that way i can study it and try to do the rest on my own.
3x+1>2andx-2<7
3x+1 > 2 and x-2 < 7
3x > 1 and x < 9
x > 1/3 AND x < 9
so x must be between 1/3 and 9
or
1/3 < x < 9
To solve the inequality 3x + 1 > 2 and x - 2 < 7, we need to solve each inequality separately and then find the overlapping solution.
First, let's solve the inequality 3x + 1 > 2:
1. Start by subtracting 1 from both sides of the inequality: 3x + 1 - 1 > 2 - 1.
This simplifies to: 3x > 1.
2. Next, divide both sides of the inequality by 3: (3x)/3 > 1/3.
This gives us: x > 1/3.
Now, let's solve the inequality x - 2 < 7:
1. Start by adding 2 to both sides of the inequality: x - 2 + 2 < 7 + 2.
This simplifies to: x < 9.
Now, we need to find the overlapping solution by combining the two inequalities. Since the first inequality (x > 1/3) requires x to be greater than 1/3, and the second inequality (x < 9) requires x to be less than 9, the overlapping solution is x ∈ (1/3, 9).
In interval notation, the solution set is (1/3, 9).