/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 Solve. $$6 x-9-x^{2} < 0$$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 35

Solve. $$6 x-9-x^{2} < 0$$

Short Answer

Expert verified
\[ (-\infty, 3) \cup (3, \infty) \].

Step by step solution

01

Rearrange the inequality

First, rewrite the inequality so that it is in a standard quadratic form. This form is \[ax^2 + bx + c < 0\]. Thus, we rearrange:\[-x^2 + 6x - 9 < 0\].To make it easier to work with, multiply the entire inequality by -1, remembering to reverse the inequality sign:\[x^2 - 6x + 9 > 0\].
02

Factor the quadratic expression

Next, factor the quadratic expression if possible. The expression \[x^2 - 6x + 9\] is a perfect square trinomial, which factors to: \[(x - 3)^2\] Now, we rewrite our inequality:\[(x - 3)^2 > 0\].
03

Solve the inequality

To solve \[(x - 3)^2 > 0\], observe that the square of a number is always non-negative:\[ (x - 3)^2 \geq 0 \].The only time \[(x - 3)^2 = 0\] is when \[x = 3\].Thus, \[(x - 3)^2 > 0\] when \[x eq 3\].
04

Write the solution

Finally, combine the results to form the solution set. Since \[(x - 3)^2 > 0\] when \[x eq 3\], the solution set is all real numbers except \[x = 3\].In interval notation, this is written as:\[ (-\infty, 3) \cup (3, \infty) \].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

inequality solution
Solving quadratic inequalities is a crucial skill in algebra. The problem we're examining involves the quadratic inequality $$ 6x-9-x^{2} < 0 $$. To solve this, we follow a series of steps that transform and simplify the inequality into a form that is easier to work with. Our goal is to find all the values of x that satisfy the inequality. We start by rewriting the inequality in standard quadratic form: $$ -x^2 + 6x - 9 < 0 $$ For ease, multiply the whole inequality by -1 (and remember to flip the inequality sign): $$ x^2 - 6x + 9 > 0 $$. Next, we focus on factoring the quadratic expression to further simplify the problem.
factoring quadratics
Factoring quadratics helps simplify the expressions, making them easier to solve. Our quadratic expression $$ x^2 - 6x + 9 $$ is a perfect square trinomial. We can factor it as: $$ (x - 3)^2 $$. So the inequality now becomes: $$ (x - 3)^2 > 0 $$. This factorization step is important because it reduces the complex quadratic expression to a simpler form.
interval notation
To express our solutions in interval notation, we need to solve the inequality $$ (x - 3)^2 > 0 $$. The expression $$ (x - 3)^2 $$ represents a square of any real number, which is always non-negative. The only time this expression equals zero is when $$ x = 3 $$. Therefore, $$ (x - 3)^2 > 0 $$ for all x except $$ x = 3 $$. In interval notation, this is represented as: $$ (-fty, 3) \cup (3, fty) $$. This notation helps to clearly depict the solution set of the inequality, excluding the particular value that does not satisfy the inequality.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Factor the polynomial function \(f(x) .\) Then solve the equation \(f(x)=0\) $$f(x)=x^{3}+2 x^{2}-13 x+10$$

Find the zeros of the function. $$g(x)=x^{4}-2 x^{3}+x^{2}-x+2[4.1]$$

List the critical values of the related function. Then solve the inequality. $$\frac{5 x}{7 x-2} > \frac{x}{x+1}$$

Factor the polynomial function \(f(x) .\) Then solve the equation \(f(x)=0\) $$f(x)=x^{3}-x^{2}-14 x+24$$

List the critical values of the related function. Then solve the inequality. $$\frac{2}{x^{2}+3} > \frac{3}{5+4 x^{2}}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.