√(2)-√(x+6) ≤-√(x)

Name: Problem 35, Chapter 7: Algebra 2
Uploaded: 2020-05-14T19:47:59-08:00
Duration: 4 min 6 s
Channel: Anna Jones
Description: √(2)-√(x+6) ≤-√(x)

Key Concepts

Domain Restrictions

When working with expressions involving square roots, it's important to define the domain of the function by ensuring that the expressions inside the square roots are non-negative. This step prevents consideration of values that are not in the domain of the original inequality and helps avoid introducing unintended solutions later in the process.

Properties of Radicals

Understanding the properties of square roots and radicals is critical when solving equations and inequalities that include them. It is important to know how to manipulate expressions with square roots, such as combining like terms or isolating radical expressions, while keeping in mind that the square root symbol typically represents the principal (non-negative) square root.

Solving Inequalities with Radicals

The process of solving inequalities that include radicals often starts with isolating the radical expressions on one side of the inequality. Once isolated, further methods, such as squaring both sides, can be applied to eliminate the radicals. However, care must be taken because squaring an inequality can potentially introduce extraneous solutions or affect the direction of the inequality if not handled correctly.

Extraneous Solutions

When both sides of an equation or inequality are squared, there is a risk of introducing solutions that satisfy the squared version but not the original problem. It is essential to verify each potential solution by substituting it back into the original inequality to ensure that it is valid within the established domain.

Transcript

00:03 All right, so right off the bat when we're looking at this problem, we see a lot of square roots, that rational symbol.

00:10 And the first thing we want to notice is this radical x over here on the right hand side has a pretty strict domain that we need to be aware of.

00:17 So all inputs need to be greater than or equal to zero because a square root can only be taken of values zero or greater.

00:27 So we have that in mind as we begin solving.

00:30 So similar to other problems, we're going to start by squaring the entire equation.

00:36 So we do want to square both sides.

00:40 On the right side, that becomes quite simple.

00:42 We're just left with x.

00:44 On the left side, however, we have a little bit of a trickier thing here.

00:48 So we'll be left with 2 plus x plus 6.

00:56 And then we're going to have, i'm just going to move this over slightly.

01:02 We have these two terms multiplied by each other.

01:05 So keep in mind that we do have to think about the distributed property here.

01:10 So we're going to have negative radical 2 radical x plus 6, and a second term that is the exact same.

01:22 So that gives us minus radical 2 radical x plus 6.

01:30 This looks pretty messy right now, but as we simplify, it won't be quite so bad...

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