Solve each quadratic inequality. Use interval notation to write each solution set. x^2-3 x ≥28

Name: Problem 55, Chapter 1: Essentials of Precalculus
Uploaded: 2021-07-09T20:36:59-08:00
Duration: 2 min 37 s
Channel: Julie Silva
Description: Solve each quadratic inequality. Use interval notation to write each solution set. x^2-3 x ≥28

Solve each quadratic inequality. Use interval notation to...

Key Concepts

Interval Testing

Once the critical points have been identified, testing the intervals between these points determines where the quadratic expression meets the inequality condition. The sign of the quadratic expression typically remains consistent within each interval, facilitating the identification of solution sets.

Quadratic Inequalities

A quadratic inequality involves a quadratic expression and an inequality sign (such as ?, ?, >, or <) instead of an equality. Solving these inequalities requires finding the set of values for the variable that make the inequality true. This concept is a core part of algebra and introduces the idea of regions or intervals on the number line where the quadratic function satisfies certain conditions.

Standard Form Conversion

Transforming the inequality into a standard form (typically by moving all terms to one side so that one side is zero) is an essential step. This process simplifies the inequality and lays the groundwork for further analysis by clearly identifying the quadratic expression that needs to be examined.

Critical Points

Critical points are the solutions to the corresponding quadratic equation (where the quadratic expression equals zero). These points are important because they divide the number line into distinct intervals. Knowing where the quadratic expression changes sign can be determined by analyzing these points.

Interval Notation

Interval notation is a concise way to represent and communicate the set of real numbers that satisfy the inequality. It uses parentheses and brackets to indicate whether endpoints are included, providing a standardized method for describing continuous ranges of solutions.

Transcript

00:01 In this problem, we're being asked to solve the given quadratic inequality.

00:03 Well, the first thing we have to do is set it equal to zero.

00:06 So i'll subtract 28 from both sets.

00:08 So we'll have x squared minus 3x minus 28 is greater than are equal to zero.

00:15 Perfect.

00:16 Okay.

00:16 Well, now that it's set to zero, we now need to find the zeros for this quadratic.

00:21 In other words, we have to set our quadratic equal to zero.

00:25 So now we'll go ahead and solve.

00:27 So i'm going to solve by factoring.

00:28 Well, there is no greatest common factor.

00:30 And the coefficient of our first term is one.

00:33 So i can set up our two binomials, and i know they'll both start with x because x times x is x squared.

00:39 So now we just need to find two numbers that multiply the negative 28 that will add to negative 3.

00:45 So i'm thinking negative 7 and positive 4.

00:48 Negative 7 plus 4 is negative 3.

00:50 Negative 7 times 4 is negative 28.

00:53 Perfect.

00:54 So we'll fill those in.

00:56 Now that we have our two factors, we'll set them both equal to 0 and then solve.

01:01 So to solve the first equation, we'll add 7 to both sides, so x could equal to 7...

Please give Ace some feedback