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

step 1 For this question, we will solve the nonlinear inequality. X squared is greater than 3 times x p

Nonlinear Inequalities Solve the nonlinear inequality....

Key Concepts

Nonlinear Inequality

A nonlinear inequality involves an expression of degree higher than one, making it not a simple linear relationship. Solving these inequalities typically involves identifying critical points where the expression equals zero, and then analyzing the intervals between these points to determine where the inequality holds true.

Quadratic Inequality

A quadratic inequality is a specific type of nonlinear inequality where the highest power of the variable is two. Solving such inequalities usually starts by rewriting the inequality in a standard form and then factoring or using the quadratic formula to determine the roots, which serve as critical values dividing the number line into regions of interest.

Factoring

Factoring is a process used to express a polynomial as a product of simpler polynomials. In the context of quadratic inequalities, factoring is used to find the roots by setting each factor equal to zero, thereby identifying the critical points that are essential for testing the intervals where the inequality is satisfied.

Sign Analysis

Sign analysis, or the test point method, is a technique where values are substituted into the factored expression across the different intervals defined by the critical points. This method helps determine whether the expression is positive or negative in each of these intervals, which is crucial for solving inequalities.

Interval Notation

Interval notation is a concise way of representing subsets of the real numbers, particularly the solution sets of inequalities. It describes intervals where the inequality holds, using parentheses and brackets to denote open and closed endpoints respectively.

Graphical Representation

Graphical representation involves plotting the solution set of an inequality on a number line. This visual tool helps illustrate which segments of the number line satisfy the inequality and indicates whether the endpoints are included or excluded based on the nature of the inequality.

Transcript

00:01 For this question, we will solve the nonlinear inequality.

00:04 X squared is greater than 3 times x plus 6, express solution in interval notation, and also graph linear equality.

00:15 So let's start by distributing 3 into x plus 6 to simplify the inequality.

00:21 We will end up with x squared is greater than 3x plus 18.

00:28 And we want to find the roots of the inequality so we can move 3x plus 18 to the left -hand side by subtracting both sides by 3x plus 18.

00:37 And we will end up with x squared minus 3x minus 18 is greater than 0.

00:46 So from here you can solve the roots by using quadratic equation or any other method to solve the roots.

00:55 I prefer the x method.

00:58 If you want to learn the x method, feel free to message me.

01:03 But as you can see, our roots are x minus 6 is equal to 0, and x plus 3 is equal to 0.

01:13 So solving for x, we get x is equal to 6, and x is equal to negative 3.

01:20 And we can plot these roots on a number line, and find numbers in between to check, see if it satisfies the inequality.

01:31 So let's start the right -hand side.

01:33 Picking a number of greater than six.

01:35 We can pick seven.

01:36 Seven squared minus three times seven minus 18, minus 18, is greater than zero.

01:44 Seven squared is 49.

01:48 Three times seven is 21, minus 18, greater than zero...

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