Question

Solve the inequality 2x - 3 < x + 2 ≤ 3x + 5. Show your work

Answer 1

The idea here is to get all the terms with an x in it together between the signs. You start with \[2x-3<x+2\le3x+5\]I am going to move all the x terms so they are in between the signs.\[-3<x+2-2x-3x \le5\]Now there is still a 2 in between the signs there from the original problem. We need to move that. Because we have to subtract it to move it from where it is currently, it will be a negative 2 on the outside. So we will put it on the left (negative numbers go on the left side of the inequality sign because x is greater than what it over there, right? x is less than what is on the outside on the right side) \[-3-2<x-2x-3x \le5\]Combining all the like terms across the board we have\[-5<-4x \le5\]Now you have to divide by -4. Divide all three "sides" by -4, but keep in mind that when you divide by a negative number you have to reverse your signs\[\frac{ 5 }{ 4 }>x \ge-\frac{ 5 }{ 4 }\]So now that it is reversed, let's rewrite it so it is in the correct form\[-\frac{ 5 }{ 4 }\le x <\frac{ 5 }{ 4 }\]See? Now the x is between the lower and the higher values. That's why you have to reverse the signs.