Describe {x : x^2+2 x<3} as an interval. Hint: Plot y=x^2+ 2 x-3

Name: Problem 25, Chapter 1: Calculus for AP
Uploaded: 2020-01-30T03:30:55-08:00
Duration: 6 min 1 s
Channel: Justin Leonesio
Description: Describe {x : x^2+2 x<3} as an interval. Hint: Plot y=x^2+ 2 x-3

Describe {x : x^2+2 x<3} as an interval. Hint: Plot y=x^2+ 2...

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, typically written in the form ax² + bx + c. They have a parabolic graph that opens upward if a is positive and downward if a is negative, with key features including the vertex, axis of symmetry, and roots. Understanding the structure of quadratic functions is essential when solving problems involving quadratic equations or inequalities.

Quadratic Inequalities

Quadratic inequalities involve expressions of the form ax² + bx + c < 0 or ax² + bx + c > 0. Solving these inequalities requires finding the roots of the corresponding quadratic equation ax² + bx + c = 0 and analyzing the sign of the expression in the intervals defined by these roots. These inequalities are used to determine the range of x values for which the quadratic expression satisfies the inequality.

Sign Analysis

Sign analysis, or the creation of a sign chart, is a method used to determine the sign of a quadratic expression across different intervals on the number line. By identifying the roots of the quadratic equation, one can partition the number line into intervals and test a value from each interval to see whether the expression is positive or negative. This technique is crucial in solving quadratic inequalities.

Interval Notation

Interval notation is a concise way to represent a set of numbers, particularly the solutions to an inequality, on the real number line. It uses parentheses and brackets to denote whether endpoints are excluded or included, respectively. This notation makes it easier to communicate solution sets and understand the relationships between different intervals.

Transcript

00:01 The goal of this problem is to express this set as an interval or an interval notation.

00:05 Let's start with inequality provided.

00:07 So we have x squared plus 2x.

00:11 We need to be less than 3.

00:14 In order to solve for all xes that make this true, it's actually easier if you allow all the terms to be on the same side.

00:20 So that instead of comparing to 3, you'll actually be comparing to 0.

00:24 Begin by subtracting 3 from both sides.

00:27 So we'll move it over here to the left along with the other terms, and we'll have then x squared, plus 2x minus 3 to now be less than 0.

00:40 And in order to solve this inequality, it actually will be really beneficial for us if we know when this expression is actually equal to 0.

00:48 Now we are technically trying to find out when it's less than 0, but when it equals 0 would be a big help to us too.

00:54 So with a quadratic expression like this, it'd be great if we could factor it.

00:59 If we can't factor, then of course we've got other methods, such as the quadratic formula or completing the square to help us, but fortunately for us, this quadratic expression does factor down very nicely into two binomials, each of which begin with an x.

01:15 In order to factor down this negative 3 as the constant term, we'll of course need a 3 and a 1, respectively.

01:22 Now, for the signs to match up perfectly, we'll need the inner part, the 3x, and the outer part, the 1x from this foil, to somehow combine together to give us a positive 2x.

01:32 And that's only possible if the 3x in the middle is positive, and the 1x on the outside is negative.

01:39 To double check, allow the positive 3 and the negative 1 to multiply, and they will indeed give us this constant term, negative 3, as part of our quadratic.

01:50 Now remember, we're checking to see when these, or essentially this expression, equals 0.

01:55 So set the parentheses that we've created equal to 0 and solve for the x value that makes that true.

02:01 In this case, you'll get a negative 3 for the x, and then for the second parentheses, you'll get a positive 1.

02:08 Okay, now these two values, again, allow the expression here, this quadratic expression on the left, to equal 0.

02:16 What they do, actually, is break apart the number line into three pieces.

02:24 And we'll visualize it here with the negative 3 on the left -hand sign and the positive 1 on the right.

02:33 Notice there's an interval over here on the left, below negative 3.

02:37 There's also an interval between negative 3 and positive 1.

02:40 Here in the middle, and then a far right interval above one.

02:44 So remember, we're checking to see when this expression here on the left is less than zero.

02:49 And these intervals are what we really need to check out now.

02:54 So in order to check out each interval, all we really need is one representative to help out.

03:00 Something less than negative three, we'll do.

03:02 Anything down there will be okay.

03:04 So we'll try out negative four.

03:07 We can plug it into either this inequality, the second one, or this third one down here.

03:12 But i would recommend plugging it into the third one.

03:15 Being that it's already factored, it will actually significantly reduce the time needed to be able to tell if it works or not with the value that we're trying...

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