Nonlinear Inequalities Solve the nonlinear inequality. Express the solution using interval notat

Name: Problem 44, Chapter 1: Algebra and Trigonometry
Uploaded: 2019-06-16T02:50:05-08:00
Duration: 4 min 9 s
Channel: Evan Vail
Description: Nonlinear Inequalities Solve the nonlinear inequality. Express the solution using interval notation, and graph the solution set. x^2+2 x>3

Nonlinear Inequalities Solve the nonlinear inequality....

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression that is set to be greater or less than a constant. Solving these inequalities typically requires finding the roots of the corresponding quadratic equation to determine intervals on which the quadratic expression is positive or negative.

Factoring

Factoring is the process of breaking down a polynomial into products of simpler polynomials or expressions. In the context of quadratic inequalities, factoring allows us to rewrite the quadratic into a product of binomials, which can then be used to identify the critical points or roots of the inequality.

Critical Points and Sign Analysis

Critical points are the x-values where the quadratic expression equals zero, and they divide the real number line into distinct intervals. By determining the sign (positive or negative) of the quadratic expression in each interval using test points, one can identify which intervals satisfy the given inequality.

Interval Notation

Interval notation is a concise way of representing a set of numbers between endpoints. It is used to denote the solution set of an inequality, where rounded brackets indicate endpoints that are not included and square brackets indicate endpoints that are included.

Graphical Representation

Graphical representation involves drawing the solution set on a number line. This typically includes marking the critical points and shading the intervals that satisfy the inequality, providing a visual interpretation of the range of values that are solutions.

Transcript

00:01 All righty.

00:02 So the first thing we have to do here for number 44 is we have to express this inequality in terms of zero.

00:10 In order before we can even find the zeros.

00:12 So we're going to subtract three from both sides.

00:14 So now that we have x squared plus 2x minus three is greater than zero, now we factor x plus three times x minus one greater than zero.

00:27 And now we define the zeros.

00:29 We just look at each one of these guys.

00:30 X equals one.

00:32 X equals negative 3.

00:34 So now we construct our little chart, which we learn how to do from the textbook around these.

00:38 Remember for the chart we go from left to right or from negative infinity to positive infinity, starting with the lowest value.

00:44 So we go negative infinity, comma, lowest one, which is negative 3.

00:51 Then we go, and remember, because this is greater than rather than greater than or equal to, we don't have to worry about brackets for any of these intervals and interval notation.

01:02 So now we go negative 3, 2 .2.

01:04 To 1 and then from 1 to infinity that way we can exclude each of these guys.

01:10 So this is all real numbers except for negative 3 and 1 express in interval notation.

01:15 So that's your key in case you forgot the specifics of doing this.

01:19 All real numbers except for the numbers which you were intending to express unless you want to do a shortcut and you would like to include brackets because you have a greater than equal to and you know that the expression for that inequality when it's expressed in terms of is always going to be defined for those zeros.

01:38 So now we include our factors.

01:41 So we have x plus 3.

01:43 We have x minus 1...

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