Problem 83
Question
Solve each equation. Identify each equation as a conditional equation, an inconsistent equation, or an identity. State the solution sets to the identities using interval notation. $$\frac{1}{x}+\frac{1}{x-1}=\frac{2 x}{x^{2}-x}$$
Step-by-Step Solution
Verified Answer
No solution; inconsistent equation.
1Step 1: Identify the common denominator
The denominators are \( x \), \( x-1 \), and \( x(x-1) \). The least common denominator (LCD) of these fractions is \( x(x-1) \).
2Step 2: Rewrite the equation with the LCD
Multiply each term by the LCD \( x(x-1) \) to eliminate the fractions: \[ x(x-1) \times \frac{1}{x} + x(x-1) \times \frac{1}{x-1} = x(x-1) \times \frac{2x}{x^2 - x} \]
3Step 3: Simplify each term
Simplify each term after multiplying by the LCD: \[ (x-1) + x = 2x \Rightarrow x - 1 + x = 2x \]
4Step 4: Solve the simplified equation
Combine like terms and solve for \( x \): \[ 2x - 1 = 2x \Rightarrow -1 = 0 \] Since this statement is false, the equation has no solution.
5Step 5: Classify the equation
Because there is no value of \( x \) that satisfies the equation, this is an inconsistent equation.
Key Concepts
solving equationsfractions in equationsleast common denominator
solving equations
When you are solving equations, you look for a value of the variable that makes the equation true. This process involves several steps:
- Identifying the problem: Understand what kind of equation you are dealing with.
- Isolating the variable: Get all variable terms on one side of the equation and all constant terms on the other.
- Simplifying the equation: Combine like terms and simplify both sides as much as possible.
In this particular problem, after identifying the least common denominator (LCD) and eliminating the fractions, we simplify the equation to get \[ 2x - 1 = 2x \]. Further simplification reveals that no solution exists, leading us to conclude that this is an inconsistent equation.
- Identifying the problem: Understand what kind of equation you are dealing with.
- Isolating the variable: Get all variable terms on one side of the equation and all constant terms on the other.
- Simplifying the equation: Combine like terms and simplify both sides as much as possible.
In this particular problem, after identifying the least common denominator (LCD) and eliminating the fractions, we simplify the equation to get \[ 2x - 1 = 2x \]. Further simplification reveals that no solution exists, leading us to conclude that this is an inconsistent equation.
fractions in equations
Dealing with fractions in equations can be tricky, but there are straightforward methods to handle them. You often need to:
- Find a common denominator: Identify the least common denominator (LCD) to combine the fractions or eliminate them.
- Multiply every term by the LCD: This step helps in eliminating the fractions and simplifies the equation.
For example, in the given problem, the fractions were \[ \frac{1}{x} + \frac{1}{x-1} = \frac{2x}{x^2 - x} \]. We found the LCD as \[ x(x-1) \] and multiplied each term by this LCD to remove the fractions. This simplifies the equation and makes it easier to solve.
- Find a common denominator: Identify the least common denominator (LCD) to combine the fractions or eliminate them.
- Multiply every term by the LCD: This step helps in eliminating the fractions and simplifies the equation.
For example, in the given problem, the fractions were \[ \frac{1}{x} + \frac{1}{x-1} = \frac{2x}{x^2 - x} \]. We found the LCD as \[ x(x-1) \] and multiplied each term by this LCD to remove the fractions. This simplifies the equation and makes it easier to solve.
least common denominator
The least common denominator (LCD) is crucial when dealing with fractions in equations. Steps to find the LCD include:
- Identifying all denominators in the problem.
- Factoring their denominators if needed.
- Finding the least common multiple of these denominators.
In our exercise, the denominators were \[ x \], \[ x-1 \], and \[ x(x-1) \]. The LCD is the least common multiple of these. Hence, we use \[ x(x-1) \], which enables us to eliminate the fractions from the equation. Multiplying all terms by this LCD simplifies our task significantly.
- Identifying all denominators in the problem.
- Factoring their denominators if needed.
- Finding the least common multiple of these denominators.
In our exercise, the denominators were \[ x \], \[ x-1 \], and \[ x(x-1) \]. The LCD is the least common multiple of these. Hence, we use \[ x(x-1) \], which enables us to eliminate the fractions from the equation. Multiplying all terms by this LCD simplifies our task significantly.
Other exercises in this chapter
Problem 82
For each pair of polynomials, use division to determine whether the first polynomial is a factor of the second. Use synthetic division when possible. If the fir
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In place of each question mark in Exercises \(75-92,\) put an expression that will make the rational expressions equivalent. $$\frac{3}{x-4}=\frac{?}{4-x}$$
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Perform the indicated operations. $$\frac{k^{2}+2 k m+m^{2}}{k^{2}-2 k m+m^{2}} \cdot \frac{m^{2}+3 m-m k-3 k}{m^{2}+m k+3 m+3 k}$$
View solution Problem 83
For each pair of polynomials, use division to determine whether the first polynomial is a factor of the second. Use synthetic division when possible. If the fir
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