Solve the inequality x-3<2(2x+1)<x+12

Answer 1

To solve this inequality, we need to distribute and simplify the terms inside the parentheses first:

x - 3 < 4x + 2 < x + 12

First, let's focus on the left inequality, x - 3 < 4x + 2:

Subtract x from both sides: -3 < 3x + 2

Subtract 2 from both sides: -5 < 3x

Divide both sides by 3, remembering that dividing by a negative number reverses the inequality:

-5/3 > x

Now let's focus on the right inequality, 4x + 2 < x + 12:

Subtract x from both sides: 3x + 2 < 12

Subtract 2 from both sides: 3x < 10

Divide both sides by 3: x < 10/3

So, the solutions for the inequality are:

-5/3 > x and x < 10/3