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

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

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

Key Concepts

Linear Inequalities

This concept involves solving inequalities in which the variable appears to the first degree. Solving linear inequalities requires similar techniques as solving linear equations, with additional attention paid to the direction of the inequality, especially when multiplying or dividing by a negative number.

Algebraic Manipulation

In solving inequalities, algebraic manipulation is essential. This includes combining like terms, distributing constants, performing addition or subtraction on both sides, and isolating the variable. It provides the necessary steps to simplify the inequality and reveal the solution set.

Fraction Operations

Working with fractions is often necessary in inequalities that involve rational expressions. This concept entails finding common denominators, combining fractional expressions, and simplifying them to make the inequality easier to solve.

Set-Builder and Interval Notation

These notations are used to express the solution sets of inequalities. Set-builder notation defines a set by specifying the properties that its members must satisfy, whereas interval notation describes continuous subsets of real numbers using endpoints, which can be open or closed depending on whether the endpoints are included in the solution.

Transcript

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

00:03 What does it mean to solve an inequality symbolically? well, given my two sides of my inequality, i want to keep it balanced so that i eventually get x on one side and everything that doesn't have x in the other.

00:14 In order to determine what values of x will always make this inequality true.

00:19 Okay, so i see this is a little more of a complex inequality.

00:22 I see fractions automatically.

00:24 I see one fraction with a denominator of four and another fraction with a denominator of two.

00:28 So i personally don't like fractions.

00:31 I'm going to try to get rid of them first.

00:33 So i'm going to look for the least common denominator.

00:35 What's the least common denominator between 4 and 2, and this x technically has a denominator of 1? well, since 2 is a factor of 4, and so is 1, i'm going to say the least common denominator is 4.

00:48 Now, why is that important? because i want to multiply by a number to both sides of the inequality in order to cancel each of the denominator.

00:55 So on the left side, i'm going to multiply by positive 4.

00:59 And on the right side, i'm also going to multiply by a positive 4.

01:02 On the left side, the 4s will cancel, and i'll just get 3x.

01:07 And then on the right side, i have to multiply each term.

01:10 Remember, a term is something that is added or subtracted.

01:13 So my first term, x times 4 is 4x, minus 2 times 4.

01:19 Okay, i can kind of cancel this.

01:21 This becomes 2.

01:22 So this will become 2 times the quantity x plus 2...