Algebra Examples

$\frac{2 x - 1}{3} - \frac{x + 13}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1

Simplify $\frac{2 x - 1}{3} - \frac{x + 13}{24}$ .

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Step 1.1

To write $\frac{2 x - 1}{3}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{2 x - 1}{3} \cdot \frac{8}{8} - \frac{x + 13}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.2

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.1

Multiply $\frac{2 x - 1}{3}$ by $\frac{8}{8}$ .

$\frac{(2 x - 1) \cdot 8}{3 \cdot 8} - \frac{x + 13}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.2.2

Multiply $3$ by $8$ .

$\frac{(2 x - 1) \cdot 8}{24} - \frac{x + 13}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{(2 x - 1) \cdot 8 - (x + 13)}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Apply the distributive property.

$\frac{2 x \cdot 8 - 1 \cdot 8 - (x + 13)}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.2

Multiply $8$ by $2$ .

$\frac{16 x - 1 \cdot 8 - (x + 13)}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.3

Multiply $- 1$ by $8$ .

$\frac{16 x - 8 - (x + 13)}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.4

Apply the distributive property.

$\frac{16 x - 8 - x - 1 \cdot 13}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.5

Multiply $- 1$ by $13$ .

$\frac{16 x - 8 - x - 13}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.6

Subtract $x$ from $16 x$ .

$\frac{15 x - 8 - 13}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.7

Subtract $13$ from $- 8$ .

$\frac{15 x - 21}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.8

Factor $3$ out of $15 x - 21$ .

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Step 1.4.8.1

Factor $3$ out of $15 x$ .

$\frac{3 (5 x) - 21}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.8.2

Factor $3$ out of $- 21$ .

$\frac{3 (5 x) + 3 (- 7)}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.4.8.3

Factor $3$ out of $3 (5 x) + 3 (- 7)$ .

$\frac{3 (5 x - 7)}{24} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.5

Cancel the common factors.

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Step 1.5.1

Factor $3$ out of $24$ .

$\frac{3 (5 x - 7)}{3 \cdot 8} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.5.2

Cancel the common factor.

$\frac{3 (5 x - 7)}{3 \cdot 8} = 3 x + \frac{5 (x + 1)}{8}$

Step 1.5.3

Rewrite the expression.

$\frac{5 x - 7}{8} = 3 x + \frac{5 (x + 1)}{8}$

Step 2

Simplify $3 x + \frac{5 (x + 1)}{8}$ .

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Step 2.1

To write $3 x$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{5 x - 7}{8} = 3 x \cdot \frac{8}{8} + \frac{5 (x + 1)}{8}$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $3 x$ and $\frac{8}{8}$ .

$\frac{5 x - 7}{8} = \frac{3 x \cdot 8}{8} + \frac{5 (x + 1)}{8}$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{5 x - 7}{8} = \frac{3 x \cdot 8 + 5 (x + 1)}{8}$

Step 2.3

Simplify the numerator.

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Step 2.3.1

Multiply $8$ by $3$ .

$\frac{5 x - 7}{8} = \frac{24 x + 5 (x + 1)}{8}$

Step 2.3.2

Apply the distributive property.

$\frac{5 x - 7}{8} = \frac{24 x + 5 x + 5 \cdot 1}{8}$

Step 2.3.3

Multiply $5$ by $1$ .

$\frac{5 x - 7}{8} = \frac{24 x + 5 x + 5}{8}$

Step 2.3.4

Add $24 x$ and $5 x$ .

$\frac{5 x - 7}{8} = \frac{29 x + 5}{8}$

Step 3

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

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Step 3.1

Subtract $\frac{29 x + 5}{8}$ from both sides of the equation.

$\frac{5 x - 7}{8} - \frac{29 x + 5}{8} = 0$

Step 3.2

Combine the numerators over the common denominator.

$\frac{5 x - 7 - (29 x + 5)}{8} = 0$

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

$\frac{5 x - 7 - (29 x) - 1 \cdot 5}{8} = 0$

Step 3.3.2

Multiply $29$ by $- 1$ .

$\frac{5 x - 7 - 29 x - 1 \cdot 5}{8} = 0$

Step 3.3.3

Multiply $- 1$ by $5$ .

$\frac{5 x - 7 - 29 x - 5}{8} = 0$

Step 3.4

Subtract $29 x$ from $5 x$ .

$\frac{- 24 x - 7 - 5}{8} = 0$

Step 3.5

Subtract $5$ from $- 7$ .

$\frac{- 24 x - 12}{8} = 0$

Step 3.6

Cancel the common factor of $- 24 x - 12$ and $8$ .

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Step 3.6.1

Factor $4$ out of $- 24 x$ .

$\frac{4 (- 6 x) - 12}{8} = 0$

Step 3.6.2

Factor $4$ out of $- 12$ .

$\frac{4 (- 6 x) + 4 (- 3)}{8} = 0$

Step 3.6.3

Factor $4$ out of $4 (- 6 x) + 4 (- 3)$ .

$\frac{4 (- 6 x - 3)}{8} = 0$

Step 3.6.4

Cancel the common factors.

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Step 3.6.4.1

Factor $4$ out of $8$ .

$\frac{4 (- 6 x - 3)}{4 (2)} = 0$

Step 3.6.4.2

Cancel the common factor.

$\frac{4 (- 6 x - 3)}{4 \cdot 2} = 0$

Step 3.6.4.3

Rewrite the expression.

$\frac{- 6 x - 3}{2} = 0$

Step 3.7

Factor $3$ out of $- 6 x - 3$ .

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Step 3.7.1

Factor $3$ out of $- 6 x$ .

$\frac{3 (- 2 x) - 3}{2} = 0$

Step 3.7.2

Factor $3$ out of $- 3$ .

$\frac{3 (- 2 x) + 3 (- 1)}{2} = 0$

Step 3.7.3

Factor $3$ out of $3 (- 2 x) + 3 (- 1)$ .

$\frac{3 (- 2 x - 1)}{2} = 0$

Step 3.8

Factor $- 1$ out of $- 2 x$ .

$\frac{3 (- (2 x) - 1)}{2} = 0$

Step 3.9

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

$\frac{3 (- (2 x) - 1 (1))}{2} = 0$

Step 3.10

Factor $- 1$ out of $- (2 x) - 1 (1)$ .

$\frac{3 (- (2 x + 1))}{2} = 0$

Step 3.11

Rewrite $- (2 x + 1)$ as $- 1 (2 x + 1)$ .

$\frac{3 (- 1 (2 x + 1))}{2} = 0$

Step 3.12

Move the negative in front of the fraction.

$- \frac{3 (2 x + 1)}{2} = 0$

Step 4

Set the numerator equal to zero.

$3 (2 x + 1) = 0$

Step 5

Solve the equation for $x$ .

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Step 5.1

Divide each term in $3 (2 x + 1) = 0$ by $3$ and simplify.

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Step 5.1.1

Divide each term in $3 (2 x + 1) = 0$ by $3$ .

$\frac{3 (2 x + 1)}{3} = \frac{0}{3}$

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor of $3$ .

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Step 5.1.2.1.1

Cancel the common factor.

$\frac{3 (2 x + 1)}{3} = \frac{0}{3}$

Step 5.1.2.1.2

Divide $2 x + 1$ by $1$ .

$2 x + 1 = \frac{0}{3}$

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $0$ by $3$ .

$2 x + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 5.3

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 5.3.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}$

Decimal Form:

$x = - 0.5$