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

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

Key Concepts

Graphing_Solutions_on_the_Number_Line

Graphing solutions on the number line offers a visual representation of the set of numbers that satisfy an inequality. It involves marking key endpoints and using open or closed circles to indicate whether each endpoint is included in the solution set.

Interval_Notation_and_Set_Notation

Interval notation and set notation provide concise ways to represent solution sets. Interval notation uses brackets and parentheses to denote whether endpoints are included or excluded, while set notation describes the collection of numbers satisfying the conditions of the inequality.

Compound_Inequalities

Compound inequalities involve two related inequalities that must be satisfied simultaneously. They arise naturally when solving absolute value inequalities, as the single absolute value condition is split into two linear inequalities.

Absolute_Value_Inequalities

Absolute value inequalities express conditions on the magnitude of an expression without regard to its sign. They are typically transformed into compound inequalities to capture both the positive and negative cases of the inequality.

Solving_Linear_Inequalities

Solving linear inequalities involves performing algebraic operations—such as addition, subtraction, multiplication, or division—while maintaining the direction of the inequality to isolate the variable. This skill is essential in simplifying and solving the compound inequalities derived from absolute value expressions.

Transcript

00:01 All right, so here we have a very complex absolute value that's related to negative 1, or positive 1.

00:07 And so we have isolated this, so that's nice.

00:11 Once we've isolated the absolute value, we can set it equal to two possibilities.

00:14 So we're going to take the inside out, and we're going to set this equal to two possibilities.

00:19 The first possibility is that 2x plus 3, all over 3, minus 1⁄2 is, okay, is in fact less than 1.

00:29 That's what you see is what you get.

00:31 The other possibility is that 2x plus 3 over 3 minus 1 half is not less than 1, but the opposite of that.

00:43 So greater than the opposite of 1, negative 1.

00:48 So here are your two possibilities.

00:50 Now when we solve this, there's a couple easy ways to do it.

00:53 I think the easiest way to do it is to multiply the entire equation by 6.

00:57 So you multiply the entire equation by 6 because that is a multiple.

01:00 Both three and two.

01:02 And we'll give you an integer coefficients and make it a little simple by getting rid of the fraction.

01:10 So three goes into six two times.

01:13 So we get two times the quantity of two x plus three.

01:17 And then two goes into six three times.

01:19 So you get three times negative one, which is negative three is less than six times one, which is six.

01:26 Over on this side, you get three goes into six, two, two times the quantity of two x plus 3 2 goes into 6 3 so that's negative 3 and then greater than negative 6 okay now that we rewritten this we can solve it i would distribute and then combine like terms so let's distribute first so you get 2 times 2 x is 4x 2 times 3 is 6 so you get 4x plus 6 minus 3 is less than 6 now we're here we get that same 4x plus 6 minus 3 but the time it's greater than negative 6.

02:07 Okay, now you're going to combine your like terms.

02:10 So combining our like terms, we can see that, let's do this in a different color, let's do orange or something.

02:19 We can see that this is 4x and 6 minus 3 is 3.

02:25 So that 4x plus 3 is less than 6.

02:29 So over here we get the same thing.

02:31 You get 4x plus 3 is great.

02:34 Than negative 6.

02:36 Okay.

02:37 Now you want to get the 4x by themselves.

02:40 So how would you undo or get rid of let's do some light blue? positive 3.

02:46 You're going to minus 3 to both sides.

02:48 You're going to get 4x is less than 3.