Solve the nonlinear inequality. Express the solution using interval notation and graph the solut

Solve the nonlinear inequality. Express the solution using...

Key Concepts

Interval Notation

Interval notation provides a concise way of expressing sets of numbers that satisfy an inequality. It clearly denotes the start and end points of a solution set as well as whether the endpoints are included or excluded, typically using parentheses for exclusion and brackets for inclusion.

Testing Intervals with a Sign Chart

After determining the roots of the quadratic, the number line is divided into intervals. By selecting test points from each interval, you can determine the sign (positive or negative) of the expression in that interval. This method helps identify where the original inequality holds true.

Graphical Representation on the Number Line

Graphing the solution set on a number line is a visual way to represent the intervals where the inequality holds. Open circles are used to indicate endpoints that are not part of the solution if the inequality is strict, while shaded regions show the intervals of valid solutions.

Quadratic Inequalities

This concept involves solving inequalities that contain quadratic expressions. It requires understanding how the quadratic function behaves—such as identifying the regions where the function is positive or negative—and is typically solved by factoring or using the quadratic formula to find the boundaries where the expression changes sign.

Factorization of Quadratic Expressions

Factorization simplifies a quadratic expression into a product of two linear factors, which allows for easier analysis of its sign changes. By expressing the quadratic in factored form, you can determine the critical points (roots) where the function equals zero and then examine the intervals between these points.

Transcript

00:01 So i went ahead and factored this polynomial that we start with for this question and got x plus 2 x plus 3.

00:07 Make sure after you did your factor and you get what i get, the first thing we want to do is that each of these two parts equal to zero and see what we get.

00:16 And this will be the two main points of our interval.

00:18 So for this one we get negative 2, for this one we get negative 3.

00:22 So when we're making our interval table, we have negative infinity to negative 3 from negative 3 to negative 2.

00:31 And from negative 2 to infinity.

00:36 I'm going to take one number from each of these intervals and plug it in to what we have right here and see whether i get positive or negative overall.

00:45 So when i take a negative 4 for the first one, negative 4 plus 2 will give me a negative.

00:51 Negative 4 plus 3 will give me a negative.

00:53 Negative times a negative will give me a positive number.

00:57 So from negative 3 to negative 2, i'll do negative 2 and 1⁄2.

01:00 Negative 2 .5 plus 2 will give me a negative.

01:04 Negative 2 .5 plus 3 will give me a positive.

01:07 Negative times a positive will give me a negative overall.

01:11 And then from 2 to infinity, negative 2 to infinity, i'll just use 0.

01:16 0 plus 2 will give me a positive.

01:18 0 plus 3 will give me a positive...