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Chapter 4: Problem 136

Exercises \(134-136\) will help you prepare for the material covered in the next section. $$\text { Solve: } \frac{x+2}{4 x+3}=\frac{1}{x}$$

Short Answer

Expert verified
The solutions of the equation are \(x = 3\) and \(x = -1\).

Step by step solution

01

Clear the Denominator and Simplify

First, we'll multiply the entire equation by the denominators 4x + 3 and x to eliminate them. The equation becomes: x(x+2) = 4x + 3
02

Expand and Rearrange

Expand the left side of the equation: \(x^2 + 2x = 4x + 3\). Then, rearrange the equation by setting it equal to 0: \(x^2 + 2x - 4x - 3 = 0\) simplifies to \(x^2 - 2x - 3 = 0\)
03

Solve for x

This is a quadratic equation in the form \(ax^2 + bx + c = 0\). We can find the roots of the equation using the quadratic formula \(x = [-b 卤 sqrt(b^2 - 4ac)] / 2a\). Substituting a=1, b=-2 and c=-3 into the formula gives \(x = [2 卤sqrt((-2)^2 - 4*1*(-3))] / 2*1\) which simplifies to \(x = 1 卤 2\). This results in two solutions for x: x=3 and x=-1.

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Key Concepts

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

Quadratic Equations
Quadratic equations are a fundamental part of algebra, represented by the formula \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable you need to solve for. Quadratics have distinctive characteristics:
  • The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \).
  • A standard quadratic equation has two solutions, which can be real or complex numbers.
  • The solutions are found where the parabola intersects the \( x \)-axis, also known as the roots of the equation.
Understanding quadratic equations is crucial as they form the basis for many real-world applications, from physics to economics.
Clearing Denominators
When solving rational equations, one of the first steps is clearing the denominators to simplify the equation. This involves multiplying both sides of the equation by the least common denominator (LCD).
  • In the equation, \( \frac{x+2}{4x+3}=\frac{1}{x} \), the LCD is \( (4x+3)x \).
  • Multiply both sides of the equation by this LCD: \( x(x+2) = 4x + 3 \).
  • This step removes the fractions from the equation, making it easier to solve.
Clearing denominators is a valuable technique for simplifying complex rational expressions, allowing you to focus on solving the core equation.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here鈥檚 how it works:
  • \( b^2 - 4ac \) is known as the discriminant and determines the nature of the roots. A positive discriminant indicates real and distinct roots, while a zero discriminant indicates a double root.
  • Plug in the values of \( a \), \( b \), and \( c \) from your quadratic equation to find the roots.
  • For example, in the equation \( x^2 - 2x - 3 = 0 \), substitute \( a = 1 \), \( b = -2 \), \( c = -3 \) into the formula.
  • This results in two solutions for \( x \): \( x = 3 \) and \( x = -1 \).
The quadratic formula is essential for finding exact solutions when factoring is complex or impossible.
Step-by-Step Problem Solving
Solving equations step-by-step ensures clarity and accuracy in your work. Here is a structured approach:
  • Identify the type of equation (e.g., rational, linear, quadratic).
  • Clear denominators, if present, to simplify the approach to solving.
  • Rearrange the equation to standard form, if necessary, similar to \( ax^2 + bx + c = 0 \).
  • Use appropriate methods such as factoring or the quadratic formula to find solutions.
  • Always double-check the solutions by substituting them back into the original equation to verify correctness.
This methodical approach to problem-solving helps break down complex math problems into manageable parts, facilitating a deeper understanding and more reliable solutions.

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.

What question can be asked to help evaluate \(\log _{3} 81 ?\)

The bar graph indicates that the percentage of first-year college students expressing antifeminist views declined after \(1970 .\) Use this information to solve. (GRAPH CANNOT COPY). The function $$f(x)=-7.52 \ln x+53$$ models the percentage of first-year college men, \(f(x)\) expressing antifeminist views (by agreeing with the statement) \(x\) years after 1969. a. Use the function to find the percentage of first-year college men expressing antifeminist views in 2008 . Round to one decimal place. Does this function value overestimate or underestimate the percentage displayed by the graph? By how much? b. Use the function to project the percentage of first-year college men who will express antifeminist views in 2015 . Round to one decimal place.

Solve each equation. Check each proposed solution by direct substitution or with a graphing utility. $$ (\ln x)^{2}=\ln x^{2} $$
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