Algebra Examples

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

Step 1

Simplify $- \frac{3}{2} \cdot (p - 2) - 2$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- \frac{3}{2} p - \frac{3}{2} \cdot - 2 - 2 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 1.1.2

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

$- \frac{p \cdot 3}{2} - \frac{3}{2} \cdot - 2 - 2 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 1.1.3

Cancel the common factor of $2$ .

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Step 1.1.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$- \frac{p \cdot 3}{2} + \frac{- 3}{2} \cdot - 2 - 2 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 1.1.3.2

Factor $2$ out of $- 2$ .

$- \frac{p \cdot 3}{2} + \frac{- 3}{2} \cdot (2 (- 1)) - 2 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 1.1.3.3

Cancel the common factor.

$- \frac{p \cdot 3}{2} + \frac{- 3}{2} \cdot (2 \cdot - 1) - 2 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 1.1.3.4

Rewrite the expression.

$- \frac{p \cdot 3}{2} - 3 \cdot - 1 - 2 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 1.1.4

Multiply $- 3$ by $- 1$ .

$- \frac{p \cdot 3}{2} + 3 - 2 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 1.1.5

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

$- \frac{3 p}{2} + 3 - 2 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 1.2

Subtract $2$ from $3$ .

$- \frac{3 p}{2} + 1 = 2 - \frac{2}{5} \cdot (2 p + 5)$

Step 2

Simplify $2 - \frac{2}{5} \cdot (2 p + 5)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$- \frac{3 p}{2} + 1 = 2 - \frac{2}{5} (2 p) - \frac{2}{5} \cdot 5$

Step 2.1.2

Multiply $- \frac{2}{5} (2 p)$ .

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Step 2.1.2.1

Multiply $2$ by $- 1$ .

$- \frac{3 p}{2} + 1 = 2 - 2 (\frac{2}{5}) p - \frac{2}{5} \cdot 5$

Step 2.1.2.2

Combine $- 2$ and $\frac{2}{5}$ .

$- \frac{3 p}{2} + 1 = 2 + \frac{- 2 \cdot 2}{5} p - \frac{2}{5} \cdot 5$

Step 2.1.2.3

Multiply $- 2$ by $2$ .

$- \frac{3 p}{2} + 1 = 2 + \frac{- 4}{5} p - \frac{2}{5} \cdot 5$

Step 2.1.2.4

Combine $\frac{- 4}{5}$ and $p$ .

$- \frac{3 p}{2} + 1 = 2 + \frac{- 4 p}{5} - \frac{2}{5} \cdot 5$

Step 2.1.3

Cancel the common factor of $5$ .

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Step 2.1.3.1

Move the leading negative in $- \frac{2}{5}$ into the numerator.

$- \frac{3 p}{2} + 1 = 2 + \frac{- 4 p}{5} + \frac{- 2}{5} \cdot 5$

Step 2.1.3.2

Cancel the common factor.

$- \frac{3 p}{2} + 1 = 2 + \frac{- 4 p}{5} + \frac{- 2}{5} \cdot 5$

Step 2.1.3.3

Rewrite the expression.

$- \frac{3 p}{2} + 1 = 2 + \frac{- 4 p}{5} - 2$

Step 2.1.4

Move the negative in front of the fraction.

$- \frac{3 p}{2} + 1 = 2 - \frac{4 p}{5} - 2$

Step 2.2

Combine the opposite terms in $2 - \frac{4 p}{5} - 2$ .

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Step 2.2.1

Subtract $2$ from $2$ .

$- \frac{3 p}{2} + 1 = - \frac{4 p}{5} + 0$

Step 2.2.2

Add $- \frac{4 p}{5}$ and $0$ .

$- \frac{3 p}{2} + 1 = - \frac{4 p}{5}$

Step 3

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

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Step 3.1

Add $\frac{4 p}{5}$ to both sides of the equation.

$- \frac{3 p}{2} + 1 + \frac{4 p}{5} = 0$

Step 3.2

To write $- \frac{3 p}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{3 p}{2} \cdot \frac{5}{5} + \frac{4 p}{5} + 1 = 0$

Step 3.3

To write $\frac{4 p}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{3 p}{2} \cdot \frac{5}{5} + \frac{4 p}{5} \cdot \frac{2}{2} + 1 = 0$

Step 3.4

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

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Step 3.4.1

Multiply $\frac{3 p}{2}$ by $\frac{5}{5}$ .

$- \frac{3 p \cdot 5}{2 \cdot 5} + \frac{4 p}{5} \cdot \frac{2}{2} + 1 = 0$

Step 3.4.2

Multiply $2$ by $5$ .

$- \frac{3 p \cdot 5}{10} + \frac{4 p}{5} \cdot \frac{2}{2} + 1 = 0$

Step 3.4.3

Multiply $\frac{4 p}{5}$ by $\frac{2}{2}$ .

$- \frac{3 p \cdot 5}{10} + \frac{4 p \cdot 2}{5 \cdot 2} + 1 = 0$

Step 3.4.4

Multiply $5$ by $2$ .

$- \frac{3 p \cdot 5}{10} + \frac{4 p \cdot 2}{10} + 1 = 0$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- 3 p \cdot 5 + 4 p \cdot 2}{10} + 1 = 0$

Step 3.6

Simplify each term.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Factor $p$ out of $- 3 p \cdot 5 + 4 p \cdot 2$ .

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Step 3.6.1.1.1

Factor $p$ out of $- 3 p \cdot 5$ .

$\frac{p (- 3 \cdot 5) + 4 p \cdot 2}{10} + 1 = 0$

Step 3.6.1.1.2

Factor $p$ out of $4 p \cdot 2$ .

$\frac{p (- 3 \cdot 5) + p (4 \cdot 2)}{10} + 1 = 0$

Step 3.6.1.1.3

Factor $p$ out of $p (- 3 \cdot 5) + p (4 \cdot 2)$ .

$\frac{p (- 3 \cdot 5 + 4 \cdot 2)}{10} + 1 = 0$

Step 3.6.1.2

Multiply $- 3$ by $5$ .

$\frac{p (- 15 + 4 \cdot 2)}{10} + 1 = 0$

Step 3.6.1.3

Multiply $4$ by $2$ .

$\frac{p (- 15 + 8)}{10} + 1 = 0$

Step 3.6.1.4

Add $- 15$ and $8$ .

$\frac{p \cdot - 7}{10} + 1 = 0$

Step 3.6.2

Move $- 7$ to the left of $p$ .

$\frac{- 7 \cdot p}{10} + 1 = 0$

Step 3.6.3

Move the negative in front of the fraction.

$- \frac{7 p}{10} + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

$- \frac{7 p}{10} = - 1$

Step 5

Multiply both sides of the equation by $- \frac{10}{7}$ .

$- \frac{10}{7} (- \frac{7 p}{10}) = - \frac{10}{7} \cdot - 1$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $- \frac{10}{7} (- \frac{7 p}{10})$ .

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Step 6.1.1.1

Cancel the common factor of $10$ .

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Step 6.1.1.1.1

Move the leading negative in $- \frac{10}{7}$ into the numerator.

$\frac{- 10}{7} (- \frac{7 p}{10}) = - \frac{10}{7} \cdot - 1$

Step 6.1.1.1.2

Move the leading negative in $- \frac{7 p}{10}$ into the numerator.

$\frac{- 10}{7} \cdot \frac{- 7 p}{10} = - \frac{10}{7} \cdot - 1$

Step 6.1.1.1.3

Factor $10$ out of $- 10$ .

$\frac{10 (- 1)}{7} \cdot \frac{- 7 p}{10} = - \frac{10}{7} \cdot - 1$

Step 6.1.1.1.4

Cancel the common factor.

$\frac{10 \cdot - 1}{7} \cdot \frac{- 7 p}{10} = - \frac{10}{7} \cdot - 1$

Step 6.1.1.1.5

Rewrite the expression.

$\frac{- 1}{7} (- 7 p) = - \frac{10}{7} \cdot - 1$

Step 6.1.1.2

Cancel the common factor of $7$ .

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Step 6.1.1.2.1

Factor $7$ out of $- 7 p$ .

$\frac{- 1}{7} (7 (- p)) = - \frac{10}{7} \cdot - 1$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{- 1}{7} (7 (- p)) = - \frac{10}{7} \cdot - 1$

Step 6.1.1.2.3

Rewrite the expression.

$- - p = - \frac{10}{7} \cdot - 1$

Step 6.1.1.3

Multiply.

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Step 6.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 p = - \frac{10}{7} \cdot - 1$

Step 6.1.1.3.2

Multiply $p$ by $1$ .

$p = - \frac{10}{7} \cdot - 1$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- \frac{10}{7} \cdot - 1$ .

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Step 6.2.1.1

Multiply $- 1$ by $- 1$ .

$p = 1 (\frac{10}{7})$

Step 6.2.1.2

Multiply $\frac{10}{7}$ by $1$ .

$p = \frac{10}{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$p = \frac{10}{7}$

Decimal Form:

$p = 1.\overline{428571}$

Mixed Number Form:

$p = 1 \frac{3}{7}$