/*! 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 79 Solve each inequality by using t... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 79

Solve each inequality by using the method of your choice. State the solution set in interval notation and graph it. $$2 x+15

Short Answer

Expert verified
The solution set is \[(-\infty, -3) \cup (5, \infty)\].

Step by step solution

01

Write the inequality in standard form

Rearrange the inequality so that one side is equal to zero: \[2x + 15 < x^2\] becomes \[x^2 - 2x - 15 > 0\]
02

Solve the corresponding quadratic equation

Find the roots of the quadratic equation \(x^2 - 2x - 15 = 0\). For this, we use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -2\), and \(c = -15\). \[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)}\]\[x = \frac{2 \pm \sqrt{4 + 60}}{2}\]\[x = \frac{2 \pm 8}{2}\]So, the roots are \[x = 5\] and \[x = -3\].
03

Determine the intervals for testing

The roots divide the real number line into three intervals: 1. \((-\infty, -3)\)2. \((-3, 5)\)3. \((5, \infty)\)Test points from each interval to see where \(x^2 - 2x - 15\) is greater than zero.
04

Test the intervals

Choose a test point from each interval and substitute it back into the inequality \(x^2 - 2x - 15 > 0\): 1. Test \(x = -4\) (from \((-\infty, -3)\)): \((-4)^2 - 2(-4) - 15 = 16 + 8 - 15 = 9\), which is greater than zero.2. Test \(x = 0\) (from \((-3, 5)\)): \(0^2 - 2(0) - 15 = -15\), which is less than zero.3. Test \(x = 6\) (from \((5, \infty)\)): \(6^2 - 2(6) - 15 = 36 - 12 - 15 = 9\), which is greater than zero.
05

Determine the solution set in interval notation

From the test points, the inequality \(x^2 - 2x - 15 > 0\) is true for the intervals \((-\infty, -3)\) and \((5, \infty)\). Therefore, the solution set in interval notation is \[(-\infty, -3) \cup (5, \infty)\].
06

Graph the solution set

On a number line, shade the intervals \((-\infty, -3)\) and \((5, \infty)\), and place open circles at \(x = -3\) and \(x = 5\) to indicate that these points are not included in the solution set.

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.

quadratic functions
Quadratic functions are polynomials of degree two and can be written in the standard form: \(ax^2 + bx + c\). These functions generate a parabolic curve when graphed and have a variety of applications in science and engineering.
Key elements of quadratic functions include: the leading coefficient \(a\), the linear coefficient \(b\), and the constant term \(c\). The roots of the function, also called zeros or solutions, are the values of \(x\) where the function equals zero. The quadratic formula \(x = \frac{-b \ \pm \ sqrt{b^2 - 4ac}}{2a}\) helps find these roots.
    \t
  • The vertex form of a quadratic function is useful for identifying the vertex (the highest or lowest point on the graph).
    • \t
  • Quadratic functions may open upwards (\(a > 0\)) or downwards (\(a < 0\)).
inequalities
Inequalities are expressions involving the symbols \(<\), \(\le\), \(>\), and \(\ge\). Solving inequalities involves finding the set of values for the variable that make the inequality true.
When working with quadratic inequalities, the process is similar to solving quadratic equations but without finding precise equals.
Steps to solve include:
    \t
  • Rewriting the inequality in standard form.
    • \t
  • Determining the roots of the corresponding quadratic equation.
    • \t
  • Using these roots to divide the number line into intervals.
    • \t
  • Testing points within these intervals to see where the inequality holds true.
interval notation
Interval notation is a way to describe sets of numbers between two endpoints. It is commonly used in expressing the solution sets of inequalities.
Important points include:
    \t
  • Parentheses \((\)) are used to show that an endpoint is not included in the interval (open interval).
    • \t
  • Brackets \([\) are used to show that an endpoint is included (closed interval).
    • \t
  • Union symbols \(\cup\) are used to combine multiple intervals.
For instance, the solution set to our exercise \((\infty, -3) \cup (5, \infty)\) means all numbers less than -3 and all numbers greater than 5.
graphing
Graphing is a visual way to represent mathematical equations and inequalities. For quadratic inequalities, the graph typically involves a parabola.
    \t
  • First, graph the corresponding quadratic equation as a parabola.
    • \t
  • Then, determine where on the graph the inequality is satisfied (above or below the x-axis depending on the inequality).
    • \t
  • Shade this region on the graph to represent the solution set.
In our exercise, graphing \(x^2 - 2x - 15\) yields a parabola intersecting the x-axis at -3 and 5. The open circles indicate excluded points, and the shaded regions below -3 and above 5 represent the solution set of 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

A furniture maker buys foam rubber \(x\) times per year. The delivery charge is 400 dollar per purchase regardless of the amount purchased. The annual cost of storage is figured as 10,000 dollar because the more frequent the purchase, the less it costs for storage. So the annual cost of delivery and storage is given by $$C=400 x+\frac{10,000}{x}$$ a. Graph the function with a graphing calculator. b. Find the number of purchases per year that minimizes the annual cost of delivery and storage.

Concert Tickets At \(\$ 10\) per ticket, Willie Williams and the Wranglers will fill all 8000 seats in the Assembly Center. The manager knows that for every \(\$ 1\) increase in the price, 500 tickets will go unsold. a. Write the number of tickets sold \(n\) as a function of ticket price \(p\) Limill See Exercise 102 of Section 1.4 b. Write the total revenue as a function of the ticket price. Giviul Revenue is the product of \(n\) and \(p\) c. What ticket price will maximize the revenue?

Solve each problem. Economic Forecast The total annual profit (in thousands of dollars) for a department store chain is determined by \(P=90 x-6 x^{2}+0.1 x^{3},\) where \(x\) is the number of stores in the chain. The company now has 10 stores in operation and plans to pursue an aggressive expansion program. What happens to the total annual profit as the company opens more and more stores? For what values of \(x\) is profit increasing?

Find all of the real and imaginary zeros for each polynomial function. $$f(x)=119 x^{4}-5 x^{3}+214 x^{2}-10 x-48$$

Solve each polynomial inequality using the test-point method. $$x^{3}-3 x>0$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.