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

step 1 Okay, so I want to solve this inequality symbolically. What does it mean to solve it symbolicall

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

Key Concepts

Solution Set Notation

Solution sets of inequalities can be expressed using interval notation or set-builder notation. Interval notation employs parentheses for open intervals (excluding endpoints) and brackets for closed intervals (including endpoints), while set-builder notation defines the set by describing the properties that its elements satisfy.

Inequality Manipulation with Negative Multiplication or Division

When solving inequalities, it's crucial to remember that if both sides are multiplied or divided by a negative number, the direction of the inequality is reversed. Failure to reverse the inequality sign can lead to an incorrect solution.

Linear Inequalities

Linear inequalities involve expressions that are linear in nature, meaning the variable in question is raised only to the first power. Solving these requires isolating the variable using standard algebraic procedures while keeping track of the inequality sign.

Transcript

00:01 Okay, so i want to solve this inequality symbolically.

00:03 What does it mean to solve it symbolically? well, i want to perform properties of equality to both sides, such as adding a number to both sides, subtracting a number to both sides, distributed property, so on and so forth, so that i could figure out which values of x, and there are many values for an inequality, which values of x will make this true.

00:24 Okay.

00:26 So ultimately, my goal is to get x by itself on one side of the inequality, and then everything else to the other side.

00:32 Okay, so what do you think i would do first? well, i have a fraction, and i'm not a fan of fraction, so i'm going to try to get rid of this fraction.

00:38 I'm dividing the whole right side by negative three.

00:41 So if i'm dividing by negative three to move to the other side, i'm going to do the opposite operation, which is going to be multiply the whole right and left side by negative three.

00:51 Okay.

00:52 On the right side, it cancels, and then on the left side, i'm just left with negative three times the whole quantity 4x minus 1.

01:00 And i just have 3 minus x on the right side.

01:03 Okay.

01:04 Now my question is does my inequality flip here? when does my inequality flip? it flips when i multiply or divide by a negative number.

01:12 Okay, am i doing that here? yes, i multiply it by negative number to both sides.

01:16 So it's going to change from less than to greater than.

01:20 Now, what's next? well, i have negative three times the left side, so i'm going to distribute you to simplify a little more.

01:28 This will be negative 12x.

01:30 Negative three times positive one is plus three and then everything else will stay the same.

01:35 Okay.

01:36 I still want to get x to one side...

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