Problem 41
Question
Exercises \(31-50\) contain equations with variables in denominators. For each equation, a. Write the value or values of the variable that make a denominator zero. These are the restrictions on the variable. b. Keeping the restrictions in mind, solve the equation. $$ \frac{8 x}{x+1}=4-\frac{8}{x+1} $$
Step-by-Step Solution
Verified Answer
The solution \( x = -1 \) is not valid due to the restriction \( x \neq -1 \). Therefore, this equation has no solution.
1Step 1: Determine the restriction
To find the restriction, one must find the value of the variable that makes the denominator equal to zero. For the equation \( \frac{8x}{x+1} = 4 - \frac{8}{x+1} \), this occurs when \( x + 1 = 0 \) or \( x = -1 \). Therefore, the restriction is \( x \neq -1 \).
2Step 2: Simplify the equation
Having identified the restriction, we can now use it to simplify the equation. Multiply each term by \( (x + 1) \) to get rid of the fractions: \( 8x = 4(x+1) - 8 \). This simplifies to \( 8x = 4x + 4 - 8 \).
3Step 3: Solve for the variable
Now, the equation can be solved for \( x \). Simplify the equation by subtracting \( 4x \) from both sides to get \( 4x = -4 \). Further, divide each side by 4 to solve for \( x \), giving \( x = -1 \).
4Step 4: Check for restriction
Recall that the restriction is \( x \neq -1 \). The solution obtained is \( x = -1 \), which is the restriction on this equation. Therefore, this equation has no solution because the solution violates the restriction.
Key Concepts
Denominator RestrictionsRational EquationsAlgebraic Fractions
Denominator Restrictions
Understanding denominator restrictions is crucial when solving equations with algebraic fractions. The denominator of a fraction cannot be zero, as division by zero is undefined. This fundamental rule leads to the investigation of denominator restrictions.
For a given equation, identifying values that make any denominator zero is the first step. These values are excluded from the solution set and are commonly referred to as restrictions or excluded values.
In the context of the provided exercise \( \frac{8x}{x+1} = 4 - \frac{8}{x+1} \), the denominator is \(x+1\). Setting \(x+1=0\), we find \(x=-1\) is the restriction, because if \(x\) were \( -1\), we'd be dividing by zero, which is not permissible.
When solving the equation, the restriction must be constantly kept in mind. Any potential solution that violates the restriction, such as the \(x=-1\) found in the provided step-by-step solution, should be rejected, as it does not belong to the domain of the equation.
For a given equation, identifying values that make any denominator zero is the first step. These values are excluded from the solution set and are commonly referred to as restrictions or excluded values.
In the context of the provided exercise \( \frac{8x}{x+1} = 4 - \frac{8}{x+1} \), the denominator is \(x+1\). Setting \(x+1=0\), we find \(x=-1\) is the restriction, because if \(x\) were \( -1\), we'd be dividing by zero, which is not permissible.
When solving the equation, the restriction must be constantly kept in mind. Any potential solution that violates the restriction, such as the \(x=-1\) found in the provided step-by-step solution, should be rejected, as it does not belong to the domain of the equation.
Rational Equations
Rational equations contain ratios of polynomials, similar to fractions in arithmetic. Solving these requires a strategy that often involves clearing the denominators to simplify the equation to a polynomial one. To do this, we find the least common denominator (LCD) and multiply both sides of the equation by it.
This process turns the rational equation into a simpler form, often a linear or quadratic equation without denominators, which can be solved using familiar algebraic methods. However, before multiplying by the LCD, it's important to note any restrictions, as these must be honored throughout the solving process.
When solving \( \frac{8x}{x+1} = 4 - \frac{8}{x+1} \), multiplying by the LCD, \(x+1\), eliminates the denominators, leading to a simpler equation. However, as seen in the solution, the resulting answer \(x=-1\) is precisely the restricted value. Therefore, we conclude that there is no solution that satisfies the equation within its domain.
This process turns the rational equation into a simpler form, often a linear or quadratic equation without denominators, which can be solved using familiar algebraic methods. However, before multiplying by the LCD, it's important to note any restrictions, as these must be honored throughout the solving process.
When solving \( \frac{8x}{x+1} = 4 - \frac{8}{x+1} \), multiplying by the LCD, \(x+1\), eliminates the denominators, leading to a simpler equation. However, as seen in the solution, the resulting answer \(x=-1\) is precisely the restricted value. Therefore, we conclude that there is no solution that satisfies the equation within its domain.
Algebraic Fractions
Algebraic fractions feature variables in the numerator, the denominator, or both. They are manipulated using the same principles as numeric fractions but require additional care due to the presence of variables.
Concepts like finding a common denominator, simplifying, adding, subtracting, multiplying, and dividing are all applicable to algebraic fractions. The key difference is that with algebraic fractions, one must always consider the values for which the fractions are undefined (the restrictions).
For the exercise equation \( \frac{8x}{x+1} = 4 - \frac{8}{x+1} \), to get rid of the fractions, one strategy could be to find a common denominator and combine the terms. However, the approach taken in the solution efficiently eliminates fractions by multiplication, which yields a solvable linear equation without the need for further simplification.
The goal with algebraic fractions is to transform them into an equation that is easier to solve while keeping track of any restrictions that arise from the variables in the denominators. When solved correctly, the result should fall within the acceptable domain of the original equation.
Concepts like finding a common denominator, simplifying, adding, subtracting, multiplying, and dividing are all applicable to algebraic fractions. The key difference is that with algebraic fractions, one must always consider the values for which the fractions are undefined (the restrictions).
For the exercise equation \( \frac{8x}{x+1} = 4 - \frac{8}{x+1} \), to get rid of the fractions, one strategy could be to find a common denominator and combine the terms. However, the approach taken in the solution efficiently eliminates fractions by multiplication, which yields a solvable linear equation without the need for further simplification.
The goal with algebraic fractions is to transform them into an equation that is easier to solve while keeping track of any restrictions that arise from the variables in the denominators. When solved correctly, the result should fall within the acceptable domain of the original equation.
Other exercises in this chapter
Problem 41
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