Solve for x (3x-14)/5-(5x-6)/5=(4x+7)/10-5/2

$\frac{3 x - 14}{5} - \frac{5 x - 6}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Step 1

Simplify $\frac{3 x - 14}{5} - \frac{5 x - 6}{5}$ .

Combine the numerators over the common denominator.

$\frac{3 x - 14 - (5 x - 6)}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Simplify each term.

Apply the distributive property.

$\frac{3 x - 14 - (5 x) - - 6}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Multiply $5$ by $- 1$ .

$\frac{3 x - 14 - 5 x - - 6}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Multiply $- 1$ by $- 6$ .

$\frac{3 x - 14 - 5 x + 6}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Simplify terms.

Subtract $5 x$ from $3 x$ .

$\frac{- 2 x - 14 + 6}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Add $- 14$ and $6$ .

$\frac{- 2 x - 8}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Factor $2$ out of $- 2 x - 8$ .

Factor $2$ out of $- 2 x$ .

$\frac{2 (- x) - 8}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Factor $2$ out of $- 8$ .

$\frac{2 (- x) + 2 (- 4)}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Factor $2$ out of $2 (- x) + 2 (- 4)$ .

$\frac{2 (- x - 4)}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Factor $- 1$ out of $- x$ .

$\frac{2 (- (x) - 4)}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{2 (- (x) - 1 (4))}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Factor $- 1$ out of $- (x) - 1 (4)$ .

$\frac{2 (- (x + 4))}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Simplify the expression.

Rewrite $- (x + 4)$ as $- 1 (x + 4)$ .

$\frac{2 (- 1 (x + 4))}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Move the negative in front of the fraction.

$- \frac{2 (x + 4)}{5} = \frac{4 x + 7}{10} - \frac{5}{2}$

Step 2

Simplify $\frac{4 x + 7}{10} - \frac{5}{2}$ .

To write $- \frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{2 (x + 4)}{5} = \frac{4 x + 7}{10} - \frac{5}{2} \cdot \frac{5}{5}$

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{2}$ by $\frac{5}{5}$ .

$- \frac{2 (x + 4)}{5} = \frac{4 x + 7}{10} - \frac{5 \cdot 5}{2 \cdot 5}$

Multiply $2$ by $5$ .

$- \frac{2 (x + 4)}{5} = \frac{4 x + 7}{10} - \frac{5 \cdot 5}{10}$

Combine the numerators over the common denominator.

$- \frac{2 (x + 4)}{5} = \frac{4 x + 7 - 5 \cdot 5}{10}$

Simplify the numerator.

Multiply $- 5$ by $5$ .

$- \frac{2 (x + 4)}{5} = \frac{4 x + 7 - 25}{10}$

Subtract $25$ from $7$ .

$- \frac{2 (x + 4)}{5} = \frac{4 x - 18}{10}$

Factor $2$ out of $4 x - 18$ .

Factor $2$ out of $4 x$ .

$- \frac{2 (x + 4)}{5} = \frac{2 (2 x) - 18}{10}$

Factor $2$ out of $- 18$ .

$- \frac{2 (x + 4)}{5} = \frac{2 (2 x) + 2 (- 9)}{10}$

Factor $2$ out of $2 (2 x) + 2 (- 9)$ .

$- \frac{2 (x + 4)}{5} = \frac{2 (2 x - 9)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$- \frac{2 (x + 4)}{5} = \frac{2 (2 x - 9)}{2 \cdot 5}$

Cancel the common factor.

$- \frac{2 (x + 4)}{5} = \frac{2 (2 x - 9)}{2 \cdot 5}$

Rewrite the expression.

$- \frac{2 (x + 4)}{5} = \frac{2 x - 9}{5}$

Step 3

Move all terms containing $x$ to the left side of the equation.

Subtract $\frac{2 x - 9}{5}$ from both sides of the equation.

$- \frac{2 (x + 4)}{5} - \frac{2 x - 9}{5} = 0$

Combine the numerators over the common denominator.

$\frac{- 2 (x + 4) - (2 x - 9)}{5} = 0$

Simplify each term.

Apply the distributive property.

$\frac{- 2 x - 2 \cdot 4 - (2 x - 9)}{5} = 0$

Multiply $- 2$ by $4$ .

$\frac{- 2 x - 8 - (2 x - 9)}{5} = 0$

Apply the distributive property.

$\frac{- 2 x - 8 - (2 x) - - 9}{5} = 0$

Multiply $2$ by $- 1$ .

$\frac{- 2 x - 8 - 2 x - - 9}{5} = 0$

Multiply $- 1$ by $- 9$ .

$\frac{- 2 x - 8 - 2 x + 9}{5} = 0$

Subtract $2 x$ from $- 2 x$ .

$\frac{- 4 x - 8 + 9}{5} = 0$

Add $- 8$ and $9$ .

$\frac{- 4 x + 1}{5} = 0$

Factor $- 1$ out of $- 4 x$ .

$\frac{- (4 x) + 1}{5} = 0$

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (4 x) - 1 (- 1)}{5} = 0$

Factor $- 1$ out of $- (4 x) - 1 (- 1)$ .

$\frac{- (4 x - 1)}{5} = 0$

Rewrite $- (4 x - 1)$ as $- 1 (4 x - 1)$ .

$\frac{- 1 (4 x - 1)}{5} = 0$

Move the negative in front of the fraction.

$- \frac{4 x - 1}{5} = 0$

Step 4

Set the numerator equal to zero.

$4 x - 1 = 0$

Step 5

Solve the equation for $x$ .

Add $1$ to both sides of the equation.

$4 x = 1$

Divide each term in $4 x = 1$ by $4$ and simplify.

Divide each term in $4 x = 1$ by $4$ .

$\frac{4 x}{4} = \frac{1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{1}{4}$

Divide $x$ by $1$ .

$x = \frac{1}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}$

Decimal Form:

$x = 0.25$

Solve for x (3x-14)/5-(5x-6)/5=(4x+7)/10-5/2

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