Find the values of x that solve the inequality. |(x^2-4)/(x)| ≤3

Name: Problem 28, Chapter 3: Higher Level Mathematics
Uploaded: 2022-01-07T05:40:13-08:00
Duration: 3 min 32 s
Channel: Sarvesh Somasundaram
Description: Find the values of x that solve the inequality. |(x^2-4)/(x)| ≤3

Find the values of x that solve the inequality....

Key Concepts

Solving Rational Inequalities

Rational inequalities involve a fraction where both the numerator and denominator are functions, typically polynomials. Solving these inequalities requires finding the critical points by setting the numerator and denominator equal to zero, and then testing intervals between these points. Special care must be taken when the inequality involves an absolute value, as the problem must be split into different cases, and any multiplication by a variable expression may require consideration of the sign to avoid mistakes in the inequality direction.

Case Analysis in Inequalities

Case analysis refers to breaking down an inequality, especially one involving absolute values or variable denominators, into several manageable cases. When multiplying or dividing by an expression that could be positive or negative, it is often necessary to consider separate scenarios to preserve the correctness of the inequality. This systematic approach ensures that all possible conditions are taken into account, leading to a complete solution.

Absolute Value Inequalities

Absolute value inequalities involve expressions where the absolute value of an expression is compared to another number. The standard approach is to rewrite the inequality |A| ? B as a compound inequality -B ? A ? B, which then allows for solving for the variable within the bounds set by the inequality. This technique is foundational for understanding how the absolute value constrains the expression it encloses.

Domain of Rational Expressions

When working with rational expressions, it is crucial to consider the domain, which consists of all real numbers except those that cause the denominator to become zero. Ignoring domain restrictions can lead to incorrect solutions. In problems involving a rational expression, identifying and excluding these undefined points is a necessary step before further manipulation or solving the inequality.

Transcript

00:01 So for this question, we're given this absolute value inequality, and we have to basically solve this inequality and find out all the values of x which satisfy this inequality.

00:12 So what we can start with is by realizing our absolute rule, right? so the absolute rule states that if we are taking the absolute value of something that's less than a, right? a has to be bigger than zero for all values, and that's going to be true because three is bigger than zero.

00:27 And also that the absolute, so u is going to be bigger than or equal to negative a, and u is going to be less than or equal to positive a.

00:37 So this is a u.

00:40 So taking that into consideration, we could simplify this out to two separate expressions that we can solve, and that is this is going to be the first one for the positive conjecture, and this is going to be the second one for the negative version.

00:54 For the negative version.

00:57 So we have negative 3.

00:58 So now when we solve these, right, so we multiply x over there, x squared minus 4 is going to be less than equal to 3x.

01:07 And then on this side, we can do the same step, right? x squared minus 4 is going to be bigger than or equal to negative 3x.

01:16 And then from there we can put everything on one side.

01:21 So we have x squared minus 3x minus 4, less than equal to 0.

01:26 And same over here, x squared plus 3x minus 4 is bigger than equal to 0.

01:31 And then from there, we can simply realize that, first of all, x cannot be equal to zero because that means x would be zero in the denominator, and that would give us a false value.

01:50 So x has to be less than zero, and x also has to be bigger than zero.

01:55 So we have those two bounds for now.

01:58 And then when we solve this inequality, we can see that we have x over here.

02:02 We could factor this out...

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