Express your answer using set notation or interval notation. Graph the solution set. |1-2 x|-4<-

Express your answer using set notation or interval notation....

Key Concepts

Graphing on the Number Line

Graphing solutions on the number line involves marking the interval that represents the solution set and using open or closed circles to indicate whether the endpoints are included. This visual representation helps in understanding the range and limits of the solutions.

Interval Notation

Interval notation is a standardized method for representing continuous sets of numbers. It uses parentheses and brackets to denote open or closed endpoints, providing a concise way to describe solution sets for inequalities.

Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. This concept is important in various areas of mathematics because it describes magnitude without concern for sign.

Absolute Value Inequalities

Absolute value inequalities involve expressions with an absolute value and establish constraints that describe distances relative to a point. Solving these inequalities typically requires isolating the absolute value expression and then considering the definition of absolute value to split the inequality into two cases.

Transcript

00:01 Solving this absolute value inequality, the first step is to isolate the absolute value.

00:04 So we've got to get rid of this minus 4.

00:06 So you're going to add 4 to both sides.

00:08 And you're going to get the absolute value of 1 minus 2x is less than negative 1 plus 4, which is positive 3.

00:17 Once you have that isolated absolute value, we can remove the absolute value bars and set this equal to two different possibilities.

00:24 So the first possibility is that exactly what you see exists.

00:28 That 1 minus 2x is in fact less than 3.

00:31 The other possibility is the opposite of that exists.

00:34 That means that 1 minus 2x is not less than but greater than not 3, but negative 3.

00:39 It's just like if you were to take this left one that's exactly what you see is what you get and multiply it by negative 1, it would generate the one on the right.

00:49 Okay? and once you have that set up, then you can just solve it like normal.

00:55 So the first thing you're going to do is you're going to want to get rid of that plus 1, so minus one to both sides, minus one to both sides.

01:02 You can end up with negative 2x is less than 2, or negative 2x is greater than negative 4.

01:11 Then you're going to want to undo the times by negative 2.

01:15 So you're going to divide by a negative 2 across the inequality on both sides of inequality.

01:21 Remember, when you divide by a negative across an inequality, you're going to reverse the inequality so that less than becomes greater than.

01:27 So you end up with x is greater than negative one.

01:31 On the other side, reverse the inequality, you get x is less than positive 2.

01:38 So our two possibilities here are that x is less than 2 or greater than negative 1.

01:42 And notice that these two things are going towards each other.

01:45 So really it's on a problem that's in and problem.

01:48 In other words, it's only going to work where both of these are true.

01:52 Because there is overlap...

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