Express your answer using set notation or interval notation. Graph the solution set. |(2 x-3)/(2

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

Key Concepts

Absolute Value

Absolute value represents the non-negative magnitude of a number or expression regardless of its sign. It involves transforming any negative input into its positive counterpart, which is crucial for interpreting distances and deviations in various mathematical contexts.

Absolute Value Inequality

An absolute value inequality involves an expression within absolute value signs compared to a number, leading to a compound inequality. The key idea (for example, |A| > B) is to consider the separate cases A > B and A < -B, ensuring the cases address when the expression exceeds the threshold in magnitude.

Set Notation and Interval Notation

Set notation and interval notation are formal ways to describe the solution sets of equations or inequalities. They enable concise representation of all values that satisfy a condition, where interval notation uses endpoints and includes or excludes boundaries with parentheses or brackets.

Graphing on the Number Line

Graphing the solution set on the number line offers a visual summary of the intervals where the inequality holds true. It involves marking boundary points, using open or closed circles based on whether endpoints are included, and shading the regions that represent the solution.

Transcript

00:01 In this situation, you want to take the inside of the absolute values.

00:04 It's already isolated.

00:06 And you have to set it equal to two possibilities here.

00:09 So first thing is that 2x minus 3 all over 2 plus 1 third is exactly the value you expect, which means it's greater than 1.

00:18 The other possibility is that 2x minus 3 all over 2 plus 1 3 is equal to the opposite of what you thought, which means that everything on the other side of it is the opposite as well.

00:28 So we're going to change the greater than to less than, change the 1 to the opposite of 1, negative 1.

00:36 And now you're going to solve this.

00:38 I would multiply everything by 6 because it's going to cancel out both of our denominators and make a nice simple inequality relationship with integer coefficients.

00:49 So 2 goes into 6 3 times.

00:51 This is going to be 3 times the quantity of 2x minus 3.

00:55 3 goes into 6 twice.

00:57 So that's plus 2 times 1, which is 2.

00:59 It's greater than 6.

01:02 On the other side here, you're going to get 3 times 2x minus 3, if that's the same.

01:08 And then plus 2, that's the same.

01:11 But then instead of bigger than 6, it's going to be less than negative 6.

01:16 Now we're going to solve this.

01:18 So 3 times 2x is going to be 6x.

01:21 3 times negative 3 is negative 9.

01:23 So you get 6x minus 9 plus 2 is greater than 6.

01:27 And the other side you get 6x minus 9 plus 2 is less than negative 6.

01:33 Now you're going to combine your like terms.

01:36 When you combine your like terms, you're going to do this on like the orange.

01:41 You're going to just combine negative 9 and positive 2, both situations here.

01:46 And that's negative 7.

01:47 So you get 6x minus 7 is greater than 6.

01:51 And 6x minus 7 is less than negative 6.

01:55 Now you're going to want to undo the minus 7.

01:59 Seven to get the 6x by itself.

02:02 So let's do that in ground or something.

02:06 So you're going to add seven to both sides and you get 6x is greater than 13.

02:12 And over here you add 7 to both sides and you get 6x is less than positive 1.

02:22 And then you're going to want to solve.

02:24 So undo times by 6 or divide by 6 both sides.

02:29 And you get your answer that x is in fact greater than 13 6, which is 2 and 1 6.

02:39 And over here you get x is less than 1 .6.

02:43 So that's basically the gist of it.

02:47 When you're graphing this, i think it's easier to graph this if you don't necessarily have any definitive intervals because you have a pretty long graph to go from 1 6.

03:00 Well, it's not that long.

03:02 But i guess we could do it.

03:03 So it's all the values that are greater than 2 and 1 6 and less than one six.

03:10 It's actually going the opposite direction.

03:11 So this is an oral relationship.

03:14 So let's see if we can fit it.

03:15 That would be what? that's like 136.

03:18 Can we fit 136? let's see.

03:21 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

03:28 Just barely.

03:29 Let's do it.

03:30 Take this off.