Express your answer using set notation or interval notation. Graph the solution set. |1-2 x|>3

Name: Problem 49, Chapter 1: Algebra and Trigonometry
Uploaded: 2020-07-09T19:41:43-08:00
Duration: 4 min 24 s
Channel: Joshua Fischbach
Description: Express your answer using set notation or interval notation. Graph the solution set. |1-2 x|>3

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

Key Concepts

Absolute Value

The absolute value represents the distance of a number from zero on the number line, regardless of direction. In algebra, it is commonly used to express the magnitude of a quantity without considering its sign, and plays a crucial role in many equations and inequalities.

Absolute Value Inequalities

These inequalities involve conditions with absolute values and require the consideration of distances on the number line. Solving them typically involves breaking the inequality into two separate cases, one for the positive and one for the negative scenario, to capture all possible solutions that satisfy the condition.

Set Notation and Interval Notation

Set notation and interval notation are concise ways to represent the solution sets of equations and inequalities. Set notation lists the collection of elements that satisfy a condition, whereas interval notation uses endpoints and brackets or parentheses to denote continuous ranges, indicating whether endpoints are included or excluded.

Graphing on the Number Line

Graphing solution sets on a number line provides a visual representation of the intervals where the inequality holds true. This graphical method involves marking critical points, using open or closed circles to indicate whether endpoints are excluded or included, and shading the corresponding portions of the number line that form the solution.

Transcript

00:01 To solve this absolute value inequality, you're just going to set this up with two different possibilities.

00:08 So the first possibility is exactly what you see is what you get.

00:11 So when you remove the absolute value bars, you just take the inside and you leave the relationship exactly as it is.

00:20 So 1 minus 2x is greater than 3.

00:23 The other possibility is that you're going to have the opposite of that value, right? that means is that the 1 minus 2x stays the same, but you're going to change the greater than to its opposite inequality, less than, and you're going to change the value that it's related to 3 to its opposite value, negative 3.

00:45 So it's kind of like you took this equation to the left and multiplied it by a negative 1, right? and you get the equation on the right.

00:52 And then you're going to solve these two and figure out the possibilities.

00:57 So the way we do that, of course, is we're going to isolate the negative 2x by subtract.

01:01 One from both sides.

01:03 So i'll do that.

01:05 You get negative 2x is greater than 2 on the left hand side.

01:10 And on the right hand side, you're going to get negative 2x is less than negative 4.

01:17 And the next step is going to be to undo times by negative 2.

01:21 So the inverse operation would be to divide by negative 2.

01:23 So you're going to divide by negative 2 across the inequality.

01:27 When you divide by a negative across an inequality, you got to remember to reverse the inequality.

01:32 So what you get is x is not greater than, but less than 2 divided by negative 2, which is negative 1.

01:38 And over here on the other side, you're going to divide by negative 2 across the inequality.

01:43 And you're going to get x is not less than, but greater than.

01:49 And then negative 4 divided by negative 2 is positive 2.

01:52 So the result that you have here is an or relationship because they're going in opposite directions.

01:57 They don't overlap.

01:58 So all the values, the less than negative 1 could work.

02:02 The values that are greater than two could work...

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