Algebra Examples

$x = - \frac{1}{2} x + \frac{1}{2} (- \frac{3}{2} x + 9) + 1$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \frac{1}{2} x + \frac{1}{2} \cdot (- \frac{3}{2} x + 9) + 1 = x$

Step 2

Simplify $- \frac{1}{2} x + \frac{1}{2} \cdot (- \frac{3}{2} x + 9) + 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine $x$ and $\frac{1}{2}$ .

$- \frac{x}{2} + \frac{1}{2} \cdot (- \frac{3}{2} x + 9) + 1 = x$

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

Combine $x$ and $\frac{3}{2}$ .

$- \frac{x}{2} + \frac{1}{2} \cdot (- \frac{x \cdot 3}{2} + 9) + 1 = x$

Step 2.1.2.2

Move $3$ to the left of $x$ .

$- \frac{x}{2} + \frac{1}{2} \cdot (- \frac{3 x}{2} + 9) + 1 = x$

Step 2.1.3

Apply the distributive property.

$- \frac{x}{2} + \frac{1}{2} (- \frac{3 x}{2}) + \frac{1}{2} \cdot 9 + 1 = x$

Step 2.1.4

Multiply $\frac{1}{2} (- \frac{3 x}{2})$ .

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Step 2.1.4.1

Multiply $\frac{1}{2}$ by $\frac{3 x}{2}$ .

$- \frac{x}{2} - \frac{3 x}{2 \cdot 2} + \frac{1}{2} \cdot 9 + 1 = x$

Step 2.1.4.2

Multiply $2$ by $2$ .

$- \frac{x}{2} - \frac{3 x}{4} + \frac{1}{2} \cdot 9 + 1 = x$

Step 2.1.5

Combine $\frac{1}{2}$ and $9$ .

$- \frac{x}{2} - \frac{3 x}{4} + \frac{9}{2} + 1 = x$

Step 2.2

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

$- \frac{x}{2} \cdot \frac{2}{2} - \frac{3 x}{4} + \frac{9}{2} + 1 = x$

Step 2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

$- \frac{x \cdot 2}{2 \cdot 2} - \frac{3 x}{4} + \frac{9}{2} + 1 = x$

Step 2.3.2

Multiply $2$ by $2$ .

$- \frac{x \cdot 2}{4} - \frac{3 x}{4} + \frac{9}{2} + 1 = x$

Step 2.4

Combine the numerators over the common denominator.

$\frac{- x \cdot 2 - 3 x}{4} + \frac{9}{2} + 1 = x$

Step 2.5

Find the common denominator.

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Step 2.5.1

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$\frac{- x \cdot 2 - 3 x}{4} + \frac{9}{2} \cdot \frac{2}{2} + 1 = x$

Step 2.5.2

Multiply $\frac{9}{2}$ by $\frac{2}{2}$ .

$\frac{- x \cdot 2 - 3 x}{4} + \frac{9 \cdot 2}{2 \cdot 2} + 1 = x$

Step 2.5.3

Write $1$ as a fraction with denominator $1$ .

$\frac{- x \cdot 2 - 3 x}{4} + \frac{9 \cdot 2}{2 \cdot 2} + \frac{1}{1} = x$

Step 2.5.4

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$\frac{- x \cdot 2 - 3 x}{4} + \frac{9 \cdot 2}{2 \cdot 2} + \frac{1}{1} \cdot \frac{4}{4} = x$

Step 2.5.5

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

$\frac{- x \cdot 2 - 3 x}{4} + \frac{9 \cdot 2}{2 \cdot 2} + \frac{4}{4} = x$

Step 2.5.6

Multiply $2$ by $2$ .

$\frac{- x \cdot 2 - 3 x}{4} + \frac{9 \cdot 2}{4} + \frac{4}{4} = x$

Step 2.6

Combine the numerators over the common denominator.

$\frac{- x \cdot 2 - 3 x + 9 \cdot 2 + 4}{4} = x$

Step 2.7

Simplify each term.

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Step 2.7.1

Multiply $2$ by $- 1$ .

$\frac{- 2 x - 3 x + 9 \cdot 2 + 4}{4} = x$

Step 2.7.2

Multiply $9$ by $2$ .

$\frac{- 2 x - 3 x + 18 + 4}{4} = x$

Step 2.8

Simplify terms.

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Step 2.8.1

Subtract $3 x$ from $- 2 x$ .

$\frac{- 5 x + 18 + 4}{4} = x$

Step 2.8.2

Add $18$ and $4$ .

$\frac{- 5 x + 22}{4} = x$

Step 2.8.3

Factor $- 1$ out of $- 5 x$ .

$\frac{- (5 x) + 22}{4} = x$

Step 2.8.4

Rewrite $22$ as $- 1 (- 22)$ .

$\frac{- (5 x) - 1 (- 22)}{4} = x$

Step 2.8.5

Factor $- 1$ out of $- (5 x) - 1 (- 22)$ .

$\frac{- (5 x - 22)}{4} = x$

Step 2.8.6

Simplify the expression.

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Step 2.8.6.1

Rewrite $- (5 x - 22)$ as $- 1 (5 x - 22)$ .

$\frac{- 1 (5 x - 22)}{4} = x$

Step 2.8.6.2

Move the negative in front of the fraction.

$- \frac{5 x - 22}{4} = x$

Step 3

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

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Step 3.1

Subtract $x$ from both sides of the equation.

$- \frac{5 x - 22}{4} - x = 0$

Step 3.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{5 x - 22}{4} - x \cdot \frac{4}{4} = 0$

Step 3.3

Combine $- x$ and $\frac{4}{4}$ .

$- \frac{5 x - 22}{4} + \frac{- x \cdot 4}{4} = 0$

Step 3.4

Combine the numerators over the common denominator.

$\frac{- (5 x - 22) - x \cdot 4}{4} = 0$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

$\frac{- (5 x) - - 22 - x \cdot 4}{4} = 0$

Step 3.5.2

Multiply $5$ by $- 1$ .

$\frac{- 5 x - - 22 - x \cdot 4}{4} = 0$

Step 3.5.3

Multiply $- 1$ by $- 22$ .

$\frac{- 5 x + 22 - x \cdot 4}{4} = 0$

Step 3.5.4

Multiply $4$ by $- 1$ .

$\frac{- 5 x + 22 - 4 x}{4} = 0$

Step 3.5.5

Subtract $4 x$ from $- 5 x$ .

$\frac{- 9 x + 22}{4} = 0$

Step 3.6

Factor $- 1$ out of $- 9 x$ .

$\frac{- (9 x) + 22}{4} = 0$

Step 3.7

Rewrite $22$ as $- 1 (- 22)$ .

$\frac{- (9 x) - 1 (- 22)}{4} = 0$

Step 3.8

Factor $- 1$ out of $- (9 x) - 1 (- 22)$ .

$\frac{- (9 x - 22)}{4} = 0$

Step 3.9

Rewrite $- (9 x - 22)$ as $- 1 (9 x - 22)$ .

$\frac{- 1 (9 x - 22)}{4} = 0$

Step 3.10

Move the negative in front of the fraction.

$- \frac{9 x - 22}{4} = 0$

Step 4

Set the numerator equal to zero.

$9 x - 22 = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Add $22$ to both sides of the equation.

$9 x = 22$

Step 5.2

Divide each term in $9 x = 22$ by $9$ and simplify.

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Step 5.2.1

Divide each term in $9 x = 22$ by $9$ .

$\frac{9 x}{9} = \frac{22}{9}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $9$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{22}{9}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{22}{9}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{22}{9}$

Decimal Form:

$x = 2.\overline{4}$

Mixed Number Form:

$x = 2 \frac{4}{9}$