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Clearing denominators is a technique used to solve rational equations by eliminating the fractions. It involves multiplying both sides of the equation by a common factor, known as the least common multiple (LCM), to eliminate the denominators.
Rational equations appear in various fields, including physics, engineering, and economics, where they are used to model real-world problems. For example, in electrical engineering, rational equations are used to analyze circuits and determine the current and voltage in a circuit. In economics, rational equations are used to model supply and demand curves.
The LCM of two or more numbers is the smallest number that is a multiple of each of the given numbers. In the context of clearing denominators, the LCM is used to eliminate the fractions in a rational equation.
When multiplying both sides of an equation by a common factor, the equation remains balanced, and the solution is not affected.
Rational expressions can be simplified by cancelling out common factors in the numerator and denominator.
Identify the rational equation and determine the denominators.
Find the LCM of the denominators.
Multiply both sides of the equation by the LCM to eliminate the denominators.
Simplify the rational expression by cancelling out common factors in the numerator and denominator.
Solve the resulting equation.
Solve the rational equation: $\frac{x}{2} + \frac{3}{4} = \frac{1}{2}$
$$\frac{x}{2} + \frac{3}{4} = \frac{1}{2}$$ Multiply both sides by 4 to eliminate the denominators: $$2x + 3 = 2$$ Subtract 3 from both sides: $$2x = -1$$ Divide both sides by 2: $$x = -\frac{1}{2}$$
$$x = -\frac{1}{2}$$
Solve the rational equation: $\frac{x}{3} - \frac{2}{5} = \frac{1}{15}$
$$\frac{x}{3} - \frac{2}{5} = \frac{1}{15}$$ Multiply both sides by 15 to eliminate the denominators: $$5x - 6 = 1$$ Add 6 to both sides: $$5x = 7$$ Divide both sides by 5: $$x = \frac{7}{5}$$
$$x = \frac{7}{5}$$
Solve the rational equation: $\frac{x}{4} + \frac{1}{2} = \frac{1}{4}$
$$\frac{x}{4} + \frac{1}{2} = \frac{1}{4}$$ Multiply both sides by 4 to eliminate the denominators: $$x + 2 = 1$$ Subtract 2 from both sides: $$x = -1$$
$$x = -1$$
Failing to find the LCM can lead to incorrect solutions.
Failing to multiply both sides of the equation by the LCM can lead to incorrect solutions.
Failing to simplify the rational expression can lead to incorrect solutions.
Always check your work by plugging the solution back into the original equation.
Using a table to compare methods can help identify the most efficient approach.
Practicing solving rational equations will help build confidence and fluency.
Graphing calculators can be used to visualize rational equations and find solutions.
Statistical software can be used to solve rational equations and analyze data.
Symbolic math tools can be used to solve rational equations and simplify expressions.
Rational equations are used to analyze circuits and determine the current and voltage in a circuit.
Rational equations are used to model supply and demand curves and determine the equilibrium price and quantity.
Rational equations are used to model the motion of objects and determine the velocity and acceleration.
What is the least common multiple (LCM) of 4 and 6?
A) 2 B) 4 C) 6 D) 12
D) 12
The LCM of 4 and 6 is 12.
A) 2 is a factor of both 4 and 6, but it is not the LCM. B) 4 is a factor of 4, but it is not the LCM. C) 6 is a factor of 6, but it is not the LCM.
What is the solution to the rational equation $\frac{x}{2} + \frac{3}{4} = \frac{1}{2}$?
A) $x = -\frac{1}{2}$ B) $x = \frac{1}{2}$ C) $x = \frac{3}{2}$ D) $x = -\frac{3}{2}$
A) $x = -\frac{1}{2}$
The solution to the rational equation is $x = -\frac{1}{2}$.
B) $x = \frac{1}{2}$ is a plausible solution, but it is not the correct solution. C) $x = \frac{3}{2}$ is not a solution to the rational equation. D) $x = -\frac{3}{2}$ is not a solution to the rational equation.
What is the LCM of 3 and 5?
A) 3 B) 5 C) 6 D) 15
D) 15
The LCM of 3 and 5 is 15.
A) 3 is a factor of 3, but it is not the LCM. B) 5 is a factor of 5, but it is not the LCM. C) 6 is not a factor of 3 or 5.
Systems of equations involve multiple equations with multiple variables. Solving systems of equations requires using techniques such as substitution and elimination.
Inequalities involve comparing two or more expressions using greater than or less than symbols. Solving inequalities requires using techniques such as adding or subtracting the same value to both sides.
Graphing rational functions involves using techniques such as factoring and canceling to simplify the function and then using the x-intercepts and y-intercepts to graph the function.
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