Solve the given inequality, giving the solution set as an interval or union of intervals. x^3 >

step 1 In this problem, we're being asked to solve the given inequality. Well, because we're dealing wi

Solve the given inequality, giving the solution set as an...

Key Concepts

Polynomial Inequalities

A polynomial inequality involves finding the set of values for the variable that makes a polynomial expression either greater than, less than, or equal to zero. The process typically begins by rewriting the inequality in standard form, factoring the polynomial, and then using various techniques to determine where the polynomial assumes the required sign.

Factoring Polynomials

Factoring is the method of decomposing a polynomial into a product of simpler polynomials. This serves as a crucial step in solving inequalities because factoring reveals the zeros or critical points of the polynomial, which are used to divide the number line into intervals for sign analysis.

Sign Analysis

Sign analysis involves examining the signs of factors over intervals determined by the critical points. By testing values in these intervals, one can determine whether the product of the factors is positive or negative, thus identifying the regions where the original inequality holds.

Interval Notation

Interval notation is a concise way of representing subsets of the real numbers, particularly solution sets to equations or inequalities. It clearly indicates the continuous range of values for which an inequality is satisfied, including whether the endpoints are part of the solution set (closed) or not (open).

Transcript

00:01 In this problem, we're being asked to solve the given inequality.

00:03 Well, because we're dealing with a polynomial function, we first need to find its zeros, meaning we're going to, instead of having our inequality sign, we'll put our equal sign.

00:11 All right, well, now, to solve this, i'm going to solve by factoring.

00:14 So first, we'll set it equal to zero by subtracting 4x from both sides.

00:18 So up x to the 3rd minus 4x equal to 0.

00:22 Now that it's set equal to 0, we can factor.

00:24 So first, i'll factor out the x, the greatest common factor, which leaves us with x squared minus 4, and x squared minus 4, that's an example of the difference of two squares.

00:33 So we can set up our two binomial factors.

00:36 The squared of x squared is x, so they both start with x.

00:39 The square of the 4 is 2, they both end with 2.

00:41 And to get that minus sign, one's factors plus and the other is minus.

00:45 So now we just have to set each of our factors equal to 0.

00:49 So x could equal to 0, x plus 2 could be 0, or x minus 2 could be 0.

00:54 And now we just solve.

00:55 So to solve the second equation, we subtract 2 from both sides.

00:59 X can be negative 2.

01:00 And to solve the last, we add two to both sides.

01:03 So we get x is equal to positive 2.

01:05 So now we have each of our zeros.