Question

Solve................................​

Answer 1

$\frac{x}{4} - \frac{x}{3} = \frac{x - 2}{9} \\ \\ \frac{x \times 3}{4 \times 3} - \frac{x \times 4}{3 \times 4} = \frac{x - 2}{9} \\ \\ \frac{3x}{12} - \frac{4x}{12} = \frac{x - 2}{9} \\ \\ \frac{3x - 4x}{12} = \frac{x - 2}{9} \\ \\ \frac{ - 1x}{12} = \frac{x - 2}{9} \\ \\ -1 x \times 9 = (x - 2) \times 12 \\ \\ - 9x = 12x - 24 \\ \\ 24 - 9x = 12x \\ \\ 24 = 12x + 9x \\ \\ 24 = 21x \\ \\ x = \frac{24}{21} \\ \\ x = 1.143$

The value of x is 1.143

Answer 2

$\rm{\displaystyle \frac{x}{4}-\frac{x}{3}=\frac{x-2}{9}}$

In order to factor an integer, we need to repeatedly divide it by the ascending sequence of primes. The number of times that each prime divides the original integer becomes its exponent in the final result.

$\to\rm{\displaystyle\frac{x}{2^{2}}-\frac{x}{3}=\frac{x-2}{9}}$

We need to add fractions.  The following rule is applied :

$\rm{\displaystyle\frac{A}{B}+\frac{C}{D}=\frac{\frac{L C D}{B} A+\frac{L C D}{D} C}{L C D}}$

• This example involves 2 terms. The LCD is equal to : 2² × 3

$\to\rm{\displaystyle\frac{3 x+4(-x)}{2^{2} \cdot 3}=\frac{x-2}{9}}$

We need to get rid of parentheses in this term.  All the negative factors will change sign.  In our example, we have only one negative factor.  The sign of the term will change, since there is an odd number of negative factors.

$\to\rm{\displaystyle\frac{3 x-4 x}{2^{2} \cdot 3}=\frac{x-2}{9}}$

We need to combine like terms in this expression by adding up all numerical coefficients and copying the literal part, if any.  No numerical coefficient implies value of 1.

$\to\rm{\displaystyle-\frac{x}{2^{2} \cdot 3}=\frac{x-2}{9}}$

In order to factor an integer, we need to repeatedly divide it by the ascending sequence of primes. The number of times that each prime divides the original integer becomes its exponent in the final result.

$\to\rm{\displaystyle-\frac{x}{2^{2} \cdot 3}=\frac{x-2}{3^{2}}}$

We need to get rid of all the denominators in this equation.  This can be achieved by multiplying both the left and the right side by the Least Common Denominator. In our example, the LCD is equal to 2² × 3²

$\to\rm{\displaystyle3(-x)=4(x-2)}$

We need to get rid of parentheses in this term.  All the negative factors will change sign.  In our example, we have only one negative factor.  The sign of the term will change, since there is an odd number of negative factors.

$\to\rm{\displaystyle-3 x=4(x-2)}$

We need to expand this term by multiplying a term and an expression.  

The following product distributive property will be used  : a(b + c) = ab + ac

$\to\rm{\displaystyle-3 x=4 x-4 \cdot 2}$

Numerical factors in this term have been multiplied.

$\to\rm{\displaystyle-3 x=4 x-8}$

In order to solve this linear equation, we need to group all the variable terms on one side, and all the constant terms on the other side of the equation.  In our example,  - term 4x, will be moved to the left side.  Notice that a term changes sign when it 'moves' from one side of the equation to the other.

$\to\rm{\displaystyle-3 x+(-4 x)=(4 x-8)+(-4 x)}$

We need to combine like terms in this expression by adding up all numerical coefficients and copying the literal part, if any.  No numerical coefficient implies value of 1.

$\to\rm{\displaystyle-7 x=(4 x-8)-4 x}$

We need to get rid of expression parentheses.  If there is a negative sign in front of it, each term within the expression changes sign.  Otherwise, the expression remains unchanged.  In our example, there are no negative expressions.

$\to\rm{\displaystyle -7 x=4 x-8-4 x}$

We need to organize this expression into groups of like terms, so we can combine them easier.

$\to\rm{\displaystyle -7 x=4 x-4 x-8}$

We need to combine like terms in this expression by adding up all numerical coefficients and copying the literal part, if any.  No numerical coefficient implies value of 1.

$\to\rm{\displaystyle-7 x=-8}$

In order to isolate the variable in this linear equation, we need to get rid of the coefficient that multiplies it.  This can be accomplished if both sides are divided by - 7.

$\to\rm{\displaystyle\frac{7 x}{7}=\frac{8}{7}}$

We need to reduce this fraction to the lowest terms. This can be done by dividing out those factors that appear both in the numerator and in the denominator.

$\boxed{\boxed{\to\rm{\displaystyle x=\frac{8}{7}}}}$

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$\underline{\underline{\bf{Verification}}}$

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