Chapter 11: Problem 27
Equations with Unknown in Denominator. \(4+\frac{1}{x+3}=8\)
Short Answer
Expert verified
The value of x in the equation is \(x = -\frac{11}{4}\).
Step by step solution
01
Isolate the variable term
Begin by subtracting 4 from both sides of the equation to isolate the fraction on one side. This yields \( \frac{1}{x+3} = 8 - 4 \).
02
Simplify the right side of the equation
Perform the subtraction on the right side to simplify the equation to \( \frac{1}{x+3} = 4 \).
03
Write the equation as a proportion
Express the equation as a proportion to find x. The equation \( \frac{1}{x+3} = \frac{4}{1} \) can be solved using cross multiplication.
04
Cross multiply
Multiply each side of the equation by its diagonal counterpart, which results in: \(1 \times 1 = 4 \times (x+3)\).
05
Solve for x
Simplify the equation to \(1 = 4x + 12\) by distributing the 4, then subtract 12 from both sides to isolate the variable term: \(1 - 12 = 4x \). Finally, divide by 4 to solve for x: \(\frac{-11}{4} = x\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Rational Equations
Rational equations are equations where variables appear in the denominator. Solving these requires a clear method to avoid common mistakes. The solution process involves isolating the variable and clearing the denominators to classify the problem into a more familiar form—a linear equation.
When dealing with a rational equation like \(4+\frac{1}{x+3}=8\), the goal is to find the value of x that satisfies the equation. The process starts by simplifying the equation into a form that allows the variable to be solved directly. This often involves moving terms across the equals sign to get all variable terms on one side and constants on the other, as illustrated in the provided step-by-step solution. Understanding how to handle these equations can unlock a wide range of mathematical problems and applications.
When dealing with a rational equation like \(4+\frac{1}{x+3}=8\), the goal is to find the value of x that satisfies the equation. The process starts by simplifying the equation into a form that allows the variable to be solved directly. This often involves moving terms across the equals sign to get all variable terms on one side and constants on the other, as illustrated in the provided step-by-step solution. Understanding how to handle these equations can unlock a wide range of mathematical problems and applications.
Cross Multiplication
Cross multiplication is a go-to technique for solving equations that involve proportions, including rational equations. To cross multiply, you take two fractions set equal to each other and multiply diagonally across the equal sign. Essentially, you multiply the numerator of one side by the denominator of the other side.
In the example \(\frac{1}{x+3} = \frac{4}{1}\), cross multiplication simplifies to solving \(\frac{1 \times 1}{1 \times (x+3)}\), or 1 = 4(x+3). Cross multiplication is useful because it quickly eliminates fractions, making the rest of the algebra more straightforward. It's important to be careful to multiply both sides of the equation equally to maintain the balance that is central to solving any equation.
In the example \(\frac{1}{x+3} = \frac{4}{1}\), cross multiplication simplifies to solving \(\frac{1 \times 1}{1 \times (x+3)}\), or 1 = 4(x+3). Cross multiplication is useful because it quickly eliminates fractions, making the rest of the algebra more straightforward. It's important to be careful to multiply both sides of the equation equally to maintain the balance that is central to solving any equation.
Isolating Variables
Isolating the variable is a fundamental concept in algebra. It means rearranging the equation so that the variable you are trying to solve for is on one side by itself. To do this, you'll often perform a series of inverse operations, such as adding, subtracting, multiplying, or dividing both sides of the equation, always maintaining balance.
In our example, after cross multiplying, you get \(\frac{4x+12}{1} = 1\). The next step involves isolating the variable x. You do this by getting rid of the constant, in this case, subtracting 12 from both sides, and then dividing by the coefficient of x, which is 4. Successfully isolating the variable will lead to an answer for x, which needs to be checked to ensure it doesn't make any denominator zero—otherwise, it would not be a valid solution.
In our example, after cross multiplying, you get \(\frac{4x+12}{1} = 1\). The next step involves isolating the variable x. You do this by getting rid of the constant, in this case, subtracting 12 from both sides, and then dividing by the coefficient of x, which is 4. Successfully isolating the variable will lead to an answer for x, which needs to be checked to ensure it doesn't make any denominator zero—otherwise, it would not be a valid solution.
Proportional Equations
Proportional equations state that two ratios are equal, and they can often be solved by using cross multiplication. A proportion is expressed as two fractions set equal to each other, such as \(\frac{1}{x+3} = \frac{4}{1}\).
Solving proportional equations can involve finding a common denominator, cross-multiplying, or using other methods to simplify the equation down to basic algebra. The key is to remember that proportional equations represent equality of ratios, and preserving this equality throughout the process is essential for finding the correct solution. With practice, solving proportional equations becomes a powerful tool in your algebra toolkit.
Solving proportional equations can involve finding a common denominator, cross-multiplying, or using other methods to simplify the equation down to basic algebra. The key is to remember that proportional equations represent equality of ratios, and preserving this equality throughout the process is essential for finding the correct solution. With practice, solving proportional equations becomes a powerful tool in your algebra toolkit.
