/*! 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 92 Find all real solutions of the e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 92

Find all real solutions of the equation. $$\frac{x}{2 x+7}-\frac{x+1}{x+3}=1$$

Short Answer

Expert verified
The real solutions are \(x = -\frac{7}{3}\) and \(x = -4\).

Step by step solution

01

Find a Common Denominator

To solve the equation, start by finding a common denominator to combine the fractions on the left side of the equation. The denominators are \(2x+7\) and \(x+3\). The common denominator will be their product: \((2x+7)(x+3)\).
02

Create Equivalent Fractions

Rewrite each fraction with the common denominator. The first fraction \(\frac{x}{2x+7}\) becomes \(\frac{x(x+3)}{(2x+7)(x+3)}\), and the second fraction \(\frac{x+1}{x+3}\) becomes \(\frac{(x+1)(2x+7)}{(2x+7)(x+3)}\).
03

Subtract the Fractions

Now that the fractions have a common denominator, subtract them: \[\frac{x(x+3) - (x+1)(2x+7)}{(2x+7)(x+3)} = 1\] Combine the numerators: \[x^2 + 3x - (2x^2 + 7x + 2x +7)\] which simplifies to \(-x^2 - 6x - 7\).
04

Set Up the New Equation

The equation turns into: \[\frac{x^2 + 3x - 2x^2 - 9x - 7}{(2x+7)(x+3)} = 1\] This simplifies to: \[\frac{-x^2 - 6x - 7}{(2x+7)(x+3)} = 1\].
05

Clear the Fraction

Multiply both sides by the denominator to eliminate the fraction:\[-x^2 - 6x - 7 = (2x+7)(x+3)\].
06

Expand and Simplify

Expand \((2x+7)(x+3)\) to get \(2x^2 + 6x + 7x + 21 = 2x^2 + 13x + 21\).Equating: \[-x^2 - 6x - 7 = 2x^2 + 13x + 21\].
07

Solve the Quadratic Equation

Combine like terms:\[0 = 3x^2 + 19x + 28\].Use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where \(a = 3\), \(b = 19\), \(c = 28\).
08

Calculate the Roots Using Quadratic Formula

Compute the discriminant: \[b^2 - 4ac = 19^2 - 4 \times 3 \times 28 = 361 - 336 = 25\].So, \[x = \frac{-19 \pm \sqrt{25}}{6}\].The roots are: \[x = \frac{-19 + 5}{6} = \frac{-14}{6} = -\frac{7}{3}\]and\[x = \frac{-19 - 5}{6} = \frac{-24}{6} = -4\].
09

Verify the Solutions

Check each solution in the original equation to make sure they don't make the denominator zero. For \(x = -\frac{7}{3}\), the denominators \(2x+7\) and \(x+3\) are not zero, and for \(x = -4\), neither denominator is zero.

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.

Common Denominator
When solving equations with fractions, finding a common denominator is key for simplifying the problem. In this exercise, the challenge was to handle the fractions on the left side of the equation:\[\frac{x}{2x+7} - \frac{x+1}{x+3} = 1.\]To make this subtraction manageable, both fractions must share the same denominator. Without a common denominator, it's like trying to subtract apples from oranges!**Steps to Find a Common Denominator:**
  • Identify the denominators of each fraction. In this case, they are 2x + 7 and x + 3.
  • The common denominator becomes the product of these denominators: (2x+7)(x+3).
  • Rewrite each fraction so that they share this common denominator. This involves adjusting the numerators accordingly to reflect this denominator.
This process allows us to combine the fractions into one, paving the way to solving the equation. It's a vital step in manipulating equations involving rational expressions.
Quadratic Formula
The quadratic formula is a powerful tool used to find solutions to quadratic equations of the form:\[ax^2 + bx + c = 0.\]In our exercise, after subtracting the fractions and clearing them from the equation, we arrive at a quadratic equation:\[3x^2 + 19x + 28 = 0.\]**Steps to Use the Quadratic Formula:**
  • Identify the coefficients from the equation: a = 3, b = 19, and c = 28.
  • Plug these numbers into the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • Calculate the discriminant \(b^2 - 4ac\). For our equation, it's 25.
  • Solve for the two possible values of x by substituting the discriminant back into the formula.
By using the quadratic formula, we efficiently find the solutions to the quadratic equation, even if they are complex or irrational. It's a reliable companion when tackling quadratic problems.
Real Solutions
Determining whether the solutions to a quadratic equation are real or not primarily involves calculating the discriminant:\[\Delta = b^2 - 4ac.\]In this context, the discriminant helps us understand the nature of the roots for our quadratic equation:\[3x^2 + 19x + 28 = 0.\]**Understanding the Discriminant:**
  • When \(\Delta > 0\), there are two distinct real solutions.
  • When \(\Delta = 0\), there is one real double root.
  • When \(\Delta < 0\), no real solutions exist; the roots are complex numbers.
For our problem, the discriminant was calculated as 25, which is greater than zero. This indicates two distinct real solutions for the equation:
  • \(x = -\frac{7}{3}\)
  • \(x = -4\)
Each of these solutions was checked to ensure they do not make any denominator in the original equation equal zero. This crucial check guarantees these are valid real solutions for the equation given in the exercise.

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

Shrinkage in Concrete Beams As concrete dries, it shrinks - the higher the water content, the greater the shrinkage. If a concrete beam has a water content of \(\bar{w} \mathrm{kg} / \mathrm{m}^{3},\) then it will shrink by a factor $$S=\frac{0.032 w-2.5}{10,000}$$ where \(S\) is the fraction of the original beam length that disappears due to shrinkage. (a) A beam \(12.025 \mathrm{m}\) long is cast in concrete that contains \(250 \mathrm{kg} / \mathrm{m}^{3}\) water. What is the shrinkage factor \(S ?\) How Iong will the beam be when it has dried? (b) A beam is \(10.014 \mathrm{m}\) long when wet. We want it to shrink to \(10.009 \mathrm{m},\) so the shrinkage factor should be \(S=0.00050 .\) What water content will provide this amount of shrinkage? PICTURE CANT COPY

Simplify the expression. (a) \(\left(\frac{x^{3 / 2}}{y^{-1 / 2}}\right)^{4}\left(\frac{x^{-2}}{y^{3}}\right)\) (b) \(\left(\frac{4 y^{3} z^{2 / 3}}{x^{1 / 2}}\right)^{2}\left(\frac{x^{-3} y^{6}}{8 z^{4}}\right)^{1 / 3}\)

Simplify the expression. (a) \(\left(8 a^{6} b^{3 / 2}\right)^{2 / 3}\) (b) \(\left(4 a^{6} b^{8}\right)^{3 / 2}\)

Plot the points \(P(0,3), Q(2,2),\) and \(R(5,3)\) on a coordinate plane. Where should the point \(S\) be located so that the figure \(P Q R S\) is a parallelogram? Write a brief description of the steps you took and your reasons for taking them.

Radicals Simplify the expression, and eliminate any negative exponents(s). Assume that all letters denote positive numbers. (a) \(\sqrt[4]{b^{3}} \sqrt{b}\) (b) \((2 \sqrt{a})(\sqrt[3]{a^{2}})\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

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.