Algebra Examples

1 - \frac{x - 3}{2} = \frac{2 - x}{3} + 4

$1 - \frac{x - 3}{2} = \frac{2 - x}{3} + 4$

Step 1

Simplify

1 - \frac{x - 3}{2}

$1 - \frac{x - 3}{2}$ .

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Step 1.1

Combine into one fraction.

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Step 1.1.1

Write

1

$1$ as a fraction with a common denominator.

\frac{2}{2} - \frac{x - 3}{2} = \frac{2 - x}{3} + 4

$\frac{2}{2} - \frac{x - 3}{2} = \frac{2 - x}{3} + 4$

Step 1.1.2

Combine the numerators over the common denominator.

\frac{2 - (x - 3)}{2} = \frac{2 - x}{3} + 4

$\frac{2 - (x - 3)}{2} = \frac{2 - x}{3} + 4$

\frac{2 - (x - 3)}{2} = \frac{2 - x}{3} + 4

$\frac{2 - (x - 3)}{2} = \frac{2 - x}{3} + 4$

Step 1.2

Simplify the numerator.

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Step 1.2.1

Apply the distributive property.

\frac{2 - x - - 3}{2} = \frac{2 - x}{3} + 4

$\frac{2 - x - - 3}{2} = \frac{2 - x}{3} + 4$

Step 1.2.2

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

\frac{2 - x + 3}{2} = \frac{2 - x}{3} + 4

$\frac{2 - x + 3}{2} = \frac{2 - x}{3} + 4$

Step 1.2.3

Add

2

$2$ and

3

$3$ .

\frac{- x + 5}{2} = \frac{2 - x}{3} + 4

$\frac{- x + 5}{2} = \frac{2 - x}{3} + 4$

\frac{- x + 5}{2} = \frac{2 - x}{3} + 4

$\frac{- x + 5}{2} = \frac{2 - x}{3} + 4$

Step 1.3

Simplify with factoring out.

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Step 1.3.1

Factor

- 1

$- 1$ out of

- x

$- x$ .

\frac{- (x) + 5}{2} = \frac{2 - x}{3} + 4

$\frac{- (x) + 5}{2} = \frac{2 - x}{3} + 4$

Step 1.3.2

Rewrite

5

$5$ as

- 1 (- 5)

$- 1 (- 5)$ .

\frac{- (x) - 1 (- 5)}{2} = \frac{2 - x}{3} + 4

$\frac{- (x) - 1 (- 5)}{2} = \frac{2 - x}{3} + 4$

Step 1.3.3

Factor

- 1

$- 1$ out of

- (x) - 1 (- 5)

$- (x) - 1 (- 5)$ .

\frac{- (x - 5)}{2} = \frac{2 - x}{3} + 4

$\frac{- (x - 5)}{2} = \frac{2 - x}{3} + 4$

Step 1.3.4

Simplify the expression.

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Step 1.3.4.1

Rewrite

- (x - 5)

$- (x - 5)$ as

- 1 (x - 5)

$- 1 (x - 5)$ .

\frac{- 1 (x - 5)}{2} = \frac{2 - x}{3} + 4

$\frac{- 1 (x - 5)}{2} = \frac{2 - x}{3} + 4$

Step 1.3.4.2

Move the negative in front of the fraction.

- \frac{x - 5}{2} = \frac{2 - x}{3} + 4

$- \frac{x - 5}{2} = \frac{2 - x}{3} + 4$

- \frac{x - 5}{2} = \frac{2 - x}{3} + 4

$- \frac{x - 5}{2} = \frac{2 - x}{3} + 4$

- \frac{x - 5}{2} = \frac{2 - x}{3} + 4

$- \frac{x - 5}{2} = \frac{2 - x}{3} + 4$

- \frac{x - 5}{2} = \frac{2 - x}{3} + 4

$- \frac{x - 5}{2} = \frac{2 - x}{3} + 4$

Step 2

Simplify

\frac{2 - x}{3} + 4

$\frac{2 - x}{3} + 4$ .

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Step 2.1

To write

4

$4$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

- \frac{x - 5}{2} = \frac{2 - x}{3} + 4 \cdot \frac{3}{3}

$- \frac{x - 5}{2} = \frac{2 - x}{3} + 4 \cdot \frac{3}{3}$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine

4

$4$ and

\frac{3}{3}

$\frac{3}{3}$ .

- \frac{x - 5}{2} = \frac{2 - x}{3} + \frac{4 \cdot 3}{3}

$- \frac{x - 5}{2} = \frac{2 - x}{3} + \frac{4 \cdot 3}{3}$

Step 2.2.2

Combine the numerators over the common denominator.

- \frac{x - 5}{2} = \frac{2 - x + 4 \cdot 3}{3}

$- \frac{x - 5}{2} = \frac{2 - x + 4 \cdot 3}{3}$

- \frac{x - 5}{2} = \frac{2 - x + 4 \cdot 3}{3}

$- \frac{x - 5}{2} = \frac{2 - x + 4 \cdot 3}{3}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Multiply

$4$ by

$3$ .

$- \frac{x - 5}{2} = \frac{2 - x + 12}{3}$

Step 2.3.2

Add

$2$ and

$12$ .

$- \frac{x - 5}{2} = \frac{- x + 14}{3}$

Step 2.4

Simplify with factoring out.

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Step 2.4.1

Factor

$- 1$ out of

$- x$ .

$- \frac{x - 5}{2} = \frac{- (x) + 14}{3}$

Step 2.4.2

Rewrite

$14$ as

$- 1 (- 14)$ .

$- \frac{x - 5}{2} = \frac{- (x) - 1 (- 14)}{3}$

Step 2.4.3

Factor

$- 1$ out of

$- (x) - 1 (- 14)$ .

$- \frac{x - 5}{2} = \frac{- (x - 14)}{3}$

Step 2.4.4

Simplify the expression.

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Step 2.4.4.1

Rewrite

$- (x - 14)$ as

$- 1 (x - 14)$ .

$- \frac{x - 5}{2} = \frac{- 1 (x - 14)}{3}$

Step 2.4.4.2

Move the negative in front of the fraction.

$- \frac{x - 5}{2} = - \frac{x - 14}{3}$

Step 3

Move all terms containing

$x$ to the left side of the equation.

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Step 3.1

Add

$\frac{x - 14}{3}$ to both sides of the equation.

$- \frac{x - 5}{2} + \frac{x - 14}{3} = 0$

Step 3.2

To write

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

$\frac{3}{3}$ .

$- \frac{x - 5}{2} \cdot \frac{3}{3} + \frac{x - 14}{3} = 0$

Step 3.3

To write

$\frac{x - 14}{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$- \frac{x - 5}{2} \cdot \frac{3}{3} + \frac{x - 14}{3} \cdot \frac{2}{2} = 0$

Step 3.4

Write each expression with a common denominator of

$6$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.4.1

Multiply

$\frac{x - 5}{2}$ by

$\frac{3}{3}$ .

$- \frac{(x - 5) \cdot 3}{2 \cdot 3} + \frac{x - 14}{3} \cdot \frac{2}{2} = 0$

Step 3.4.2

Multiply

$2$ by

$3$ .

$- \frac{(x - 5) \cdot 3}{6} + \frac{x - 14}{3} \cdot \frac{2}{2} = 0$

Step 3.4.3

Multiply

$\frac{x - 14}{3}$ by

$\frac{2}{2}$ .

$- \frac{(x - 5) \cdot 3}{6} + \frac{(x - 14) \cdot 2}{3 \cdot 2} = 0$

Step 3.4.4

Multiply

$3$ by

$2$ .

$- \frac{(x - 5) \cdot 3}{6} + \frac{(x - 14) \cdot 2}{6} = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- (x - 5) \cdot 3 + (x - 14) \cdot 2}{6} = 0$

Step 3.6

Simplify the numerator.

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Step 3.6.1

Apply the distributive property.

$\frac{(- x - - 5) \cdot 3 + (x - 14) \cdot 2}{6} = 0$

Step 3.6.2

Multiply

$- 1$ by

$- 5$ .

$\frac{(- x + 5) \cdot 3 + (x - 14) \cdot 2}{6} = 0$

Step 3.6.3

Apply the distributive property.

$\frac{- x \cdot 3 + 5 \cdot 3 + (x - 14) \cdot 2}{6} = 0$

Step 3.6.4

Multiply

$3$ by

$- 1$ .

$\frac{- 3 x + 5 \cdot 3 + (x - 14) \cdot 2}{6} = 0$

Step 3.6.5

Multiply

$5$ by

$3$ .

$\frac{- 3 x + 15 + (x - 14) \cdot 2}{6} = 0$

Step 3.6.6

Apply the distributive property.

$\frac{- 3 x + 15 + x \cdot 2 - 14 \cdot 2}{6} = 0$

Step 3.6.7

Move

$2$ to the left of

$x$ .

$\frac{- 3 x + 15 + 2 \cdot x - 14 \cdot 2}{6} = 0$

Step 3.6.8

Multiply

$- 14$ by

$2$ .

$\frac{- 3 x + 15 + 2 \cdot x - 28}{6} = 0$

Step 3.6.9

Add

$- 3 x$ and

$2 x$ .

$\frac{- x + 15 - 28}{6} = 0$

Step 3.6.10

Subtract

$28$ from

$15$ .

$\frac{- x - 13}{6} = 0$

Step 3.7

Factor

$- 1$ out of

$- x$ .

$\frac{- (x) - 13}{6} = 0$

Step 3.8

Rewrite

$- 13$ as

$- 1 (13)$ .

$\frac{- (x) - 1 (13)}{6} = 0$

Step 3.9

Factor

$- 1$ out of

$- (x) - 1 (13)$ .

$\frac{- (x + 13)}{6} = 0$

Step 3.10

Rewrite

$- (x + 13)$ as

$- 1 (x + 13)$ .

$\frac{- 1 (x + 13)}{6} = 0$

Step 3.11

Move the negative in front of the fraction.

$- \frac{x + 13}{6} = 0$

Step 4

Set the numerator equal to zero.

$x + 13 = 0$

Step 5

Subtract

$13$ from both sides of the equation.

$x = - 13$