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Chapter 1: Problem 39

Solve the equation and check your solution. (Some equations have no solution.) $$ 10-\frac{13}{x}=4+\frac{5}{x} $$

Short Answer

Expert verified
The solution to the equation is \(x = 3\)

Step by step solution

01

Get rid of fractions

Multiply each side by \(x\) to remove fractions: \[10x - 13 = 4x + 5\]
02

Rearrange the equation

Rearrange the terms to gather all the terms on one side of the equation: \[10x - 4x = 13 + 5\]
03

Simplify and solve

Simplify further to solve for \(x\): \[6x = 18\] Therefore, upon simplifying, we get: \(x = 3\)
04

Check your solution

To confirm if \(x=3\) is the correct solution, substitute \(x=3\) into the original equation and see if both sides are equal: \[10 - \frac{13}{3} = 4 + \frac{5}{3}\] LHS = \(10 - \frac{13}{3} = \frac{30-13}{3} = \frac{17}{3}\); RHS = \(4 + \frac{5}{3} = \frac{12+5}{3} = \frac{17}{3}\) Since both sides are equal, the solution is correct.

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Key Concepts

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

Checking Solutions
Once you've found a potential solution to an equation, it's important to verify if it satisfies the original equation. This step is called "checking the solution".
Checking ensures that the calculated value indeed works as a solution. Here's how to do it effectively:
  • Substitute the potential solution back into the original equation.
  • Calculate both sides of the equation separately.
  • Confirm if both sides are equal. If they are equal, your solution is correct.
  • If not equal, recheck your work for errors.
In our exercise, we calculated that \(x = 3\). By substituting \(x = 3\) back into the equation \(10-\frac{13}{x} = 4+\frac{5}{x}\), and confirming both sides as \(\frac{17}{3}\), we assured that \(x=3\) is indeed a valid solution.
Fractions in Equations
Equations often include fractions, which can make them look complicated at first. But don't worry, there's a straightforward method to handle them!
The first step is to eliminate fractions to simplify the equation, making it easier to solve. We can do this by multiplying every term by the least common denominator (LCD).
  • In the exercise, the equation \(10-\frac{13}{x} = 4+\frac{5}{x}\) involves fractions \(\frac{13}{x}\) and \(\frac{5}{x}\).
  • To eliminate these fractions, we multiply the entire equation by \(x\), because it's the common denominator.
  • This results in a simpler equation: \(10x - 13 = 4x + 5\).
Creating an equation without fractions allows you to proceed more easily to algebraic manipulation and solving.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging and simplifying equations to isolate the variable you want to solve for.
This manipulation involves moving terms from one side of the equation to the other. Here's how we approached it in our solution:
  • We started with the equation: \(10x - 13 = 4x + 5\).
  • The goal is to have all terms containing \(x\) on one side and constants on the other.
  • Subtract \(4x\) from both sides, yielding \(6x = 18\).
  • Finally, divide both sides by 6 to solve for \(x\): \(x = 3\).
By following these steps, algebraic manipulation helps in systematically arriving at the solution, making sure that the variable is isolated and solved accurately.

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Most popular questions from this chapter

The average price \(B\) (in dollars) of brand name prescription drugs from 1998 to 2005 can be modeled by \(B=6.928 t-3.45, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to 1998 . Use the model to find the year in which the price of the average brand name drug prescription exceeded \(\$ 75\).

Solve the inequality and write the solution set in interval notation. \(4 x^{3}-x^{4} \geq 0\)

The daily demand \(D\) (in thousands of barrels) for refined oil in the United States from 1995 to 2005 can be modeled by \(D=276.4 t+16,656, \quad 5 \leq t \leq 15\) where \(t\) represents the year, with \(t=5\) corresponding to 1995. (a) Use the model to find the year in which the demand for U.S. oil exceeded 18 million barrels a day. (b) Use the model to predict the year in which the demand for U.S. oil will exceed 22 million barrels a day.

Solve the inequality. Then graph the solution set on the real number line. \(\frac{1}{x}<4\)

The revenue \(R\) and cost \(C\) for a product are given by \(R=x(75-0.0005 x)\) and \(C=30 x+250,000\), where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold (see figure). (a) How many units must be sold to obtain a profit of at least \(\$ 750,000 ?\) (b) The demand equation for the product is \(p=75-0.0005 x\) where \(p\) is the price per unit. What prices will produce a profit of at least \(\$ 750,000 ?\) (c) As the number of units increases, the revenue eventually decreases. After this point, at what number of units is the revenue approximately equal to the cost? How should this affect the company's decision about the level of production?
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