Describe the set of real numbers satisfying |x-3|=|x-2|+1 as a half-infinite interval.

Name: Problem 26, Chapter 1: Calculus for AP
Uploaded: 2019-08-06T04:11:13-08:00
Duration: 4 min 9 s
Channel: Aymara Gallardo
Description: Describe the set of real numbers satisfying |x-3|=|x-2|+1 as a half-infinite interval.

Describe the set of real numbers satisfying |x-3|=|x-2|+1 as...

Key Concepts

Case Analysis in Absolute Value Equations

Solving equations with absolute values often requires dividing the domain into several intervals where the expressions inside the absolute values change sign. Performing a case analysis allows the equation to be rewritten without absolute value signs, making it easier to solve.

Piecewise Definition of Absolute Value

Absolute value functions can be expressed as piecewise functions, breaking the expression into different cases based on whether the input is greater than or less than a particular value. This is crucial when solving equations involving absolute values, as it often necessitates considering multiple scenarios.

Interval Notation

Interval notation is a method used to describe subsets of the real numbers, particularly useful for representing solution sets that are continuous or extend infinitely in one direction. A half-infinite interval is one that has one finite endpoint and extends indefinitely in the other direction.

Absolute Value Function

The absolute value function measures the distance of a number from zero on the number line. It is defined such that for any real number x, |x| is non-negative, capturing the idea of magnitude irrespective of sign.

Distance Interpretation

The expression |x - a| can be interpreted as the distance between the point x and the point a on the real number line. This perspective is often used to translate absolute value equations into distance comparisons.

Transcript

00:01 For this exercise, we have to write as before the result as an interval.

00:07 So the first thing is to observe we have two absolute values.

00:12 So absolute value is an expression that looks nice, but it's hard to tell what it means when you are solving the equation.

00:22 So what we are going to do here is break this problem into four smaller problems, easier problems.

00:31 So what we do is consider cases when what is inside the this absolute value is positive and what is inside the other absolute value is also positive.

00:43 When this one is positive but this one is negative, when this one is negative, this one is positive or both of them are negative.

00:53 So that's case one, case two, case three.

00:56 The first one is telling us that x has to be bigger or equal than three, this one is telling us that x has to be bigger than two.

01:06 So something that is bigger or equal than three, and also is bigger than two, is bigger than two, is bigger than three.

01:16 Okay, so that's the interval.

01:18 In this case, it's saying that x is bigger than three, but x is smaller than two, and that's not possible.

01:25 This case is telling us that x is smaller than 3 and x is bigger than 2.

01:37 So we are between 2 and 3.

01:40 And this last case tells us that x is smaller than 3 and x is smaller or equal than 2.

01:48 So that's any x that is smaller than 2 or equal than 2.

01:53 So now we are going to solve for the case 1.

01:56 So in the case 1, both of them are positive, so we can remove the absolute values.

02:03 Like nothing happened.

02:05 And now we can cancel those x, and we have from one side negative 3, from the other side, we have negative 1, negative 2 plus 1, that's negative 1.

02:17 And that's not equal to each other.

02:19 They are not equal for any x.

02:21 So there is no solution in that case...

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