Solve for x (2x+3)/3=6/(x-1)

The problem consists of a rational equation where an algebraic fraction involving the variable x is set equal to another algebraic fraction containing x. The objective is to find the value(s) of x that make the equation true. This typically involves finding a common denominator, multiplying both sides of the equation to clear the fractions, and then solving for x using algebraic methods such as combining like terms, isolating x, and checking for any potential restrictions on the solution set (for example, x cannot be equal to values that would make any denominator zero).

$\frac{2 x + 3}{3} = \frac{6}{x - 1}$

Answer

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Solution:

Step 1

Cross-multiply the terms of the given equation $\frac{2 x + 3}{3} = \frac{6}{x - 1}$ .

Step 2

Isolate the variable $x$ .

Step 2.1

First, expand the expression $(2 x + 3) (x - 1)$ .

Step 2.1.1

Utilize the FOIL (First, Outer, Inner, Last) method to expand $(2 x + 3) (x - 1)$ .

Step 2.1.1.1

Distribute $2 x$ across $(x - 1)$ and $3$ across $(x - 1)$ , resulting in $2 x (x - 1) + 3 (x - 1) = 3 \cdot 6$ .

Step 2.1.1.2

Continue distribution to obtain $2 x \cdot x - 2 x \cdot 1 + 3 (x - 1) = 3 \cdot 6$ .

Step 2.1.1.3

Complete distribution to get $2 x^{2} - 2 x + 3 x - 3 = 3 \cdot 6$ .

Step 2.1.2

Combine like terms and simplify.

Step 2.1.2.1

Handle each term individually.

Step 2.1.2.1.1

Combine powers of $x$ by adding exponents.

Step 2.1.2.1.1.1

Rewrite $2 (x \cdot x) - 2 x + 3 x - 3 = 3 \cdot 6$ .

Step 2.1.2.1.1.2

Multiply $x$ by $x$ to get $2 x^{2} - 2 x + 3 x - 3 = 3 \cdot 6$ .

Step 2.1.2.1.2

Multiply $- 1$ by $2$ to get $2 x^{2} - 2 x + 3 x - 3 = 3 \cdot 6$ .

Step 2.1.2.1.3

Multiply $3$ by $- 1$ to get $2 x^{2} - 2 x + 3 x - 3 = 3 \cdot 6$ .

Step 2.1.2.2

Combine $- 2 x$ and $3 x$ to get $2 x^{2} + x - 3 = 3 \cdot 6$ .

Step 2.2

Calculate $3 \cdot 6$ to get $2 x^{2} + x - 3 = 18$ .

Step 2.3

Subtract $18$ from both sides to get $2 x^{2} + x - 21 = 0$ .

Step 2.4

Factor the quadratic equation.

Step 2.4.1

Identify two numbers that multiply to $2 \cdot - 21 = - 42$ and add to $1$ .

Step 2.4.1.1

Express $1 x$ as the sum of $- 6 x + 7 x$ .

Step 2.4.1.2

Expand to $2 x^{2} - 6 x + 7 x - 21 = 0$ .

Step 2.4.2

Group and factor by grouping.

Step 2.4.2.1

Group as $(2 x^{2} - 6 x) + (7 x - 21) = 0$ .

Step 2.4.2.2

Factor out the GCF from each group to get $2 x (x - 3) + 7 (x - 3) = 0$ .

Step 2.4.3

Factor out the common binomial $(x - 3)$ to get $(x - 3) (2 x + 7) = 0$ .

Step 2.5

Solve for $x$ by setting each factor equal to zero.

Step 2.5.1

Set $x - 3 = 0$ and solve for $x$ to get $x = 3$ .

Step 2.5.2

Set $2 x + 7 = 0$ and solve for $x$ .

Step 2.5.2.1

Subtract $7$ from both sides to get $2 x = - 7$ .

Step 2.5.2.2

Divide by $2$ to get $x = - \frac{7}{2}$ .

Step 2.6

Combine the solutions to get $x = 3, - \frac{7}{2}$ .

Step 3

Present the solution in various forms.

Exact Form: $x = 3, - \frac{7}{2}$ Decimal Form: $x = 3, - 3.5$ Mixed Number Form: $x = 3, - 3 \frac{1}{2}$

Knowledge Notes:

Cross-Multiplication: This technique involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal to each other. It is used to solve equations involving two fractions.
FOIL Method: A mnemonic for remembering the steps to multiply two binomials: First, Outer, Inner, Last. It is a specific case of the more general method of distribution.
Distributive Property: This property states that $a (b + c) = a b + a c$ . It is used to expand expressions and to simplify equations.
Combining Like Terms: This involves adding or subtracting terms that have the same variable raised to the same power.
Factoring: The process of breaking down an expression into a product of simpler expressions. It can be used to solve quadratic equations by setting each factor equal to zero and solving for the variable.
Quadratic Equations: These are polynomial equations of the second degree, generally in the form $a x^{2} + b x + c = 0$ . They can be solved by factoring, completing the square, or using the quadratic formula.
Factoring by Grouping: A factoring technique used when an expression has four or more terms. It involves grouping terms with common factors and then factoring out the greatest common factor from each group.
Zero Product Property: This property states that if the product of two factors is zero, then at least one of the factors must be zero. It is used to find the roots of polynomial equations.
Solving Linear Equations: The process of finding the value of the variable that makes the equation true. This often involves isolating the variable on one side of the equation using inverse operations.