/*! 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 53 Solve each inequality by using t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 53

Solve each inequality by using the graphical method. State the solution set in interval notation. $$x^{2}-2 x-3>0$$

Short Answer

Expert verified
The solution set in interval notation is \(( -\infty , -1 ) \cup ( 3, \infty ) \).

Step by step solution

01

Factor the Quadratic Expression

First, factor the quadratic expression on the left side of the inequality.\[ x^2 - 2x - 3 = (x - 3)(x + 1) \]
02

Identify the Critical Points

Set each factor equal to zero to find the critical points which will be the x-values where the expression equals zero.\[ x - 3 = 0 \Rightarrow x = 3 \]\[ x + 1 = 0 \Rightarrow x = -1 \]
03

Plot the Critical Points on a Number Line

Draw a number line and plot the critical points \( x = 3 \) and \( x = -1 \). These points divide the number line into three intervals: \((-\infty, -1)\), \((-1, 3)\) and \((3, \infty)\).
04

Test a Point in Each Interval

Choose a test point from each interval to determine if the product \((x - 3)(x + 1)\) is positive or negative in that interval.\[- For \((-\infty, -1)\), choose \(x = -2\):\] \[ (-2 - 3)(-2 + 1) = (-5)(-1) = 5 > 0 \]\[ - For \((-1, 3)\), choose \(x = 0\):\] \[ (0 - 3)(0 + 1) = (-3)(1) = -3 < 0 \]\[ - For \((3, \infty)\), choose \(x = 4\):\] \[ (4 - 3)(4 + 1) = (1)(5) = 5 > 0 \]
05

Determine the Solution Set

Since we want the product to be greater than zero, the solution consists of the intervals where the product is positive. Therefore, the solution set is\[ (-\infty, -1) \cup (3, \infty) \]

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.

Understanding the Graphical Method
The graphical method is a visual approach to solve inequalities. By graphing the quadratic function, you can determine where the function is positive or negative.
When the graph of a quadratic function is above the x-axis, the function is positive. Below the x-axis, the function is negative.
For our example inequality \(x^{2}-2 x-3>0\), we can follow a few clear steps:
  • Graph the quadratic function \(y = x^2 - 2x - 3\).
  • Identify the x-intercepts (the points where the graph crosses the x-axis).
  • Determine the intervals on the x-axis where the graph is above the x-axis.

By determining these intervals, we can find the solution set for the inequality.
Factoring Quadratics Made Simple
Factoring a quadratic expression is a key step in solving inequalities.
When we factor a quadratic, we express it as a product of binomials. For \(x^2 - 2x - 3\), we can factor it into \((x - 3)(x + 1)\).

To factor a quadratic expression, you look for two numbers that multiply to the constant term and add up to the coefficient of the linear term.
  • For the quadratic \(x^2 - 2x - 3\), we need two numbers that multiply to -3 and add up to -2.
  • These numbers are -3 and 1, hence the factorization \((x - 3)(x + 1)\).
Factoring helps to identify critical points where the quadratic expression changes sign, which is essential in intervals analysis.
Decoding Interval Notation
Interval notation is a shorthand way to describe a set of numbers or a segment of the number line.
It uses parentheses \(()\) and brackets \([]\) to show if endpoints are included or excluded.
In our solution, we divide the number line into intervals based on the critical points -1 and 3.
The three intervals are:
  • \((-fty, -1)\) - numbers less than -1.
  • \((-1, 3)\) - numbers between -1 and 3.
  • \((3, fty)\) - numbers greater than 3.
We test points from each interval to see if they satisfy the inequality.
For \(x^2 - 2x - 3 > 0\), the solution set where the product is positive is given by the intervals \((-fty, -1)\) and \((3, fty)\).
Using interval notation, we write the solution as \((-)fty, -1) \cup (3, fty)\). The union symbol \(\cup\) denotes the combination of these intervals.

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

Solve each polynomial inequality using the test-point method. $$x^{4}-19 x^{2}+90 \leq 0$$

Solve the problem. Renting a Car The cost of renting a car for one day is 39 dollar plus 30 cents per mile. Write the average cost per mile \(C\) as a function of the number of miles driven in one day \(x .\) Graph the function for \(x>0 .\) What happens to \(C\) as the number of miles gets very large?

Big Barn Mike wants to enclose a rectangular area for his rabbits alongside his large barn using 30 feet of fencing. What dimensions will maximize the area fenced if the barn is used for one side of the rectangle?

Synthetic Division Explain how synthetic division can be used to find the quotient and remainder when \(3 x^{3}+4 x^{2}+2 x-4\) is divided by \(3 x-2\)

Making a Glass Tank An architect for the Aquarium of the Americas is designing a cylindrical fish tank to hold \(1000 \mathrm{ft}^{3}\) of water. The bottom and side of the tank will be 3 -in.-thick glass and the tank will have no top. a. Write the inside surface area of the tank as a function of the inside radius and graph the function. b. Use the minimum feature of a graphing calculator to find the inside radius that minimizes that inside surface area. c. If Owens Corning will build the tank for 250 dollar per cubic foot of glass used, then what will be the cost for the minimal tank?
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

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.