Describe the set as a union of finite or infinite intervals....

Key Concepts

Union of Intervals

The union of intervals is a method for expressing a set that is composed of several distinct continuous segments on the real number line. After solving an inequality, the solution set may consist of multiple intervals which, when combined, form the complete set of solutions. This representation offers a clear and concise way to describe all the regions where the inequality holds true.

Absolute Value Inequalities

Absolute value inequalities involve expressions wrapped in an absolute value operator, which measures the distance of the expression from zero. Solving these inequalities typically requires splitting the inequality into two cases, one for the positive scenario and another for the negative, to capture all possibilities where the inequality holds. This method is fundamental when handling inequalities of the form |f(x)| > c or |f(x)| < c.

Quadratic Inequalities

Quadratic inequalities are inequalities that involve a quadratic expression. The process of solving these entails determining the roots by setting the quadratic expression equal to a specific value, and then testing the intervals between and beyond these roots to establish where the inequality is satisfied. Analyzing the sign changes around the critical points is key to accurately describe the solution set.

Transcript

00:01 Okay, so to solve this absolute value problem, the first thing that we're going to do is reduce the problem of absolute values to several smaller problems.

00:12 So let's consider one case, let's call part one, in which the things inside the absolute value are positive.

00:23 So x squared plus 2x is positive.

00:26 This can happen in two cases.

00:29 Case 1, when x is positive and x plus 2 is also positive.

00:35 That means that x is bigger than 0 and x is bigger than negative 2.

00:41 And the case 2 in which x is negative and x plus 2 is negative, meaning that x is smaller than negative 2.

00:52 Okay, so in that case, the absolute value can go away and the expression that we'll have is exactly the same.

01:02 Now, see, it's like if this was 3, the absolute value of 3 is just, so we can remove the absolute values without problem.

01:13 So this is the expression that we have now.

01:15 Now we can subtract both sides negative 2, and what we are looking is at this quadratic function and we want to know when it's positive.

01:23 Now, if we make, we take the quadratic equation and we calculate to see which ones are the roots, these ones are the roots, negative 1 plus or minus square root of 3.

01:36 And if we calculate the sign chart for this function, we'll get that until negative 1 minus square root of 3 is positive, then it's negative until negative 1, negative 1.

01:52 Plus square root of 3, and then positive after that.

01:58 So this would be the part that we are interesting, this part and this part.

02:07 So in the case 1, what we have is that x has to be, which one i painted first, x has to be in that area to make this positive.

02:24 Now, at the same time, we know that since we are in the case 1, x has to be bigger than negative 2 so we paint that and then we also know that x has to be bigger than 0 so the solution that we will obtain from here is everything that has been painted twice so that's this part okay so what we have is that the solution from this case is negative 1 plus square root of 3 to infinity now in the case 2 what we have is that we want our x to be there to satisfy this equation, this inequality, and we also want it to be less than negative 2, and we want it to be less than 0...

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