Problem

Solve for x 1/(2x)+1/x=1/15

The question is asking for the correct value of 'x' that satisfies the equation provided. Specifically, the equation is a rational equation where the variable 'x' appears in the denominators of the fractions. You will need to find the common denominator, combine the fractions, and then solve for 'x' through algebraic means, often involving multiplying both sides of the equation to get rid of the fractions and isolating 'x' on one side of the equation.

12x+1x=115

Answer

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Solution:

Step:1

Identify the least common denominator (LCD) for the fractions in the equation.

Step:1.1

To find the LCD, determine the least common multiple (LCM) of the denominators 2x, x, and 15.

Step:1.2

The LCM process involves two parts: finding the LCM of the numerical coefficients 2, 1, and 15, and then finding the LCM for the variable parts x1 and x1.

Step:1.3

The LCM is the smallest number that each of the original numbers can divide into without a remainder. The process includes:

  1. Listing out the prime factors for each number.

  2. Multiplying each prime factor by the highest power it appears with in any of the numbers.

Step:1.4

The number 2 is already prime.

Step:1.5

The number 1 is not considered prime as it only has one distinct positive divisor, itself.

Step:1.6

The number 15 has prime factors 3 and 5, or 35.

Step:1.7

The LCM of 2, 1, and 15 is found by multiplying the prime factors, each to their highest power, resulting in 235.

Step:1.8

Perform the multiplication of 235.

Step:1.8.1

First, multiply 2 by 3 to get 65.

Step:1.8.2

Then, multiply 6 by 5 to get 30.

Step:1.9

The variable x1 simplifies to x, which appears once.

Step:1.10

The LCM of x1 and x1 is simply x, as it is already the highest power of x present.

Step:1.11

Combine the numeric LCM 30 with the variable part x to get the overall LCD of 30x.

Step:2

Clear the fractions by multiplying each term in the equation 12x+1x=115 by the LCD 30x.

Step:2.1

Apply the multiplication to each term: 30x2x+30xx=30x15.

Step:2.2

Begin simplifying the left side of the equation.

Step:2.2.1

Simplify each fraction individually.

Step:2.2.1.1

Use the commutative property to rearrange the multiplication.

Step:2.2.1.2

Reduce common factors where possible.

Step:2.2.1.2.1

Extract the factor of 2 from 30.

Step:2.2.1.2.2

Extract the factor of 2 from 2x.

Step:2.2.1.2.3

Cancel out the common factors.

Step:2.2.1.2.4

Rewrite the simplified expression.

Step:2.2.1.3

Combine the coefficient 15 with 1x.

Step:2.2.1.4

Cancel out the common variable x.

Step:2.2.1.4.1

Perform the cancellation.

Step:2.2.1.4.2

Rewrite the simplified expression.

Step:2.2.1.5

Apply the commutative property again if necessary.

Step:2.2.1.6

Combine the coefficient 30 with 1x.

Step:2.2.1.7

Cancel out the common variable x.

Step:2.2.1.7.1

Perform the cancellation.

Step:2.2.1.7.2

Rewrite the simplified expression.

Step:2.2.2

Add the numbers 15 and 30 together.

Step:2.3

Simplify the right side of the equation.

Step:2.3.1

Cancel out the common factor of 15.

Step:2.3.1.1

Factor out 15 from 30x.

Step:2.3.1.2

Perform the cancellation.

Step:2.3.1.3

Rewrite the simplified expression.

Step:3

Solve the simplified equation for x.

Step:3.1

Rewrite the equation to isolate the term with x.

Step:3.2

Divide both sides of the equation by 2 to solve for x.

Step:3.2.1

Apply the division to both sides of the equation.

Step:3.2.2

Simplify the left side of the equation.

Step:3.2.2.1

Cancel out the common factor of 2.

Step:3.2.2.1.1

Perform the cancellation.

Step:3.2.2.1.2

Divide x by 1 to find the value of x.

Step:4

Present the solution in various forms.

Exact Form: x=452

Decimal Form: x=22.5

Mixed Number Form: x=2212

Knowledge Notes:

The problem involves solving a linear equation with fractions. The key steps in solving such an equation include:

  1. Finding the Least Common Denominator (LCD): This is crucial for combining fractions by ensuring that all terms have the same denominator. The LCD is the least common multiple (LCM) of the denominators.

  2. Clearing the Fractions: By multiplying each term by the LCD, we eliminate the fractions, simplifying the equation.

  3. Simplifying the Equation: This involves reducing terms, canceling common factors, and combining like terms.

  4. Solving for the Variable: Once the equation is simplified, we isolate the variable on one side to find its value.

  5. Prime Factorization: This is used to find the LCM of the numerical parts of the denominators. Prime factorization breaks down a number into its prime factors.

  6. Commutative Property of Multiplication: This property states that the order of factors can be changed without affecting the product, which can be useful for rearranging terms during simplification.

  7. Cancellation: When a number or variable appears in both the numerator and denominator, it can be canceled out, simplifying the expression.

Understanding these concepts is essential for solving equations with fractions and can be applied to a wide range of algebraic problems.

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