Solve the inequality symbolically. Express the solution set in set-builder or interval notation.

step 1 Okay, so I'm going to solve this inequality symbolically. What does mean to solve the inequality

Solve the inequality symbolically. Express the solution set...

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions that relate a linear function to a value using inequality symbols such as >, <, ?, or ?. They involve similar steps to solving linear equations such as combining like terms and isolating the variable, but require special attention to operations—especially multiplication or division by a negative number—which can change the direction of the inequality.

Set-Builder and Interval Notation

Set-builder and interval notation are methods to represent the solution sets of inequalities. Set-builder notation describes all elements that satisfy a given condition, while interval notation provides a compact way to denote continuous ranges of numbers. Both notations are essential for communicating the exact solution set of an inequality in algebra.

Transcript

00:01 Okay, so i'm going to solve this inequality symbolically.

00:03 What does mean to solve the inequality symbolically? well, my goal is to find all the solutions of x that make this inequality true, and this is going to be specifically when i solve for x, get it on one side of the inequality, and then i could express that solution in interval notation.

00:21 An interval notation is just another way to write the solution involving parentheses and brackets.

00:27 Okay.

00:27 So i see automatically that this inequality has a fraction in it, and i don't like my fraction.

00:34 I'm multiplying by one half, same thing as dividing by two to the entire right side.

00:40 So i'm going to try to get rid of that fraction.

00:42 I'm multiplying by one half, same thing as dividing by two, so i could also multiply by the reciprocal or just multiply by two to both sides.

00:53 Right and i'm multiplying by two to the entire left side.

00:58 This will cancel two in one half.

01:00 So i'm going to get two times the quantity 2x minus three.

01:08 And then on the right, i'm just going to get x plus one.

01:11 When i multiply by two, does that flip my inequality? it does not, right? recall that my inequality flips when i multiply or divide by a negative number.

01:21 Okay.

01:22 What would you do next? i still want to get x by itself, but i have, i'm multiplying 2 by this x over here, so i'm going to distribute that first.

01:30 2 times 2x is 4x, minus 2 times 3 is 6, and the rest of my inequality just stays same.

01:39 So i'm going to get x completely by itself, so i'm going to move everything that has x to one side and everything that doesn't have x to the other.

01:45 So how do i move this x to the left side? well, i just have x...

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