Finite Math Examples

$\frac{1}{2} \cdot (p + 1) < \frac{3}{4} \cdot (p - 1)$

Step 1

Simplify $\frac{1}{2} \cdot (p + 1)$ .

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

$\frac{1}{2} \cdot (p + 1) < \frac{3}{4} \cdot (p - 1)$

Step 1.3

Apply the distributive property.

$\frac{1}{2} p + \frac{1}{2} \cdot 1 < \frac{3}{4} \cdot (p - 1)$

Step 1.4

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

$\frac{p}{2} + \frac{1}{2} \cdot 1 < \frac{3}{4} \cdot (p - 1)$

Step 1.5

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

$\frac{p}{2} + \frac{1}{2} < \frac{3}{4} \cdot (p - 1)$

Step 2

Simplify $\frac{3}{4} \cdot (p - 1)$ .

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Step 2.1

Apply the distributive property.

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

Step 2.2

Combine $\frac{3}{4}$ and $p$ .

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

Step 2.3

Multiply $\frac{3}{4} \cdot - 1$ .

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Step 2.3.1

Combine $\frac{3}{4}$ and $- 1$ .

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

Step 2.3.2

Multiply $3$ by $- 1$ .

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

Step 2.4

Move the negative in front of the fraction.

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

Step 3

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

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Step 3.1

Subtract $\frac{3 p}{4}$ from both sides of the inequality.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Subtract $3 p$ from $p \cdot 2$ .

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Step 3.5.1

Reorder $p$ and $2$ .

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

Step 3.5.2

Subtract $3 p$ from $2 \cdot p$ .

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Move all terms not containing $p$ to the right side of the inequality.

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Step 4.1

Subtract $\frac{1}{2}$ from both sides of the inequality.

$- \frac{p}{4} < - \frac{3}{4} - \frac{1}{2}$

Step 4.2

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

$- \frac{p}{4} < - \frac{3}{4} - \frac{1}{2} \cdot \frac{2}{2}$

Step 4.3

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

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Step 4.3.1

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

$- \frac{p}{4} < - \frac{3}{4} - \frac{2}{2 \cdot 2}$

Step 4.3.2

Multiply $2$ by $2$ .

$- \frac{p}{4} < - \frac{3}{4} - \frac{2}{4}$

Step 4.4

Combine the numerators over the common denominator.

$- \frac{p}{4} < \frac{- 3 - 2}{4}$

Step 4.5

Subtract $2$ from $- 3$ .

$- \frac{p}{4} < \frac{- 5}{4}$

Step 4.6

Move the negative in front of the fraction.

$- \frac{p}{4} < - \frac{5}{4}$

Step 5

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- p < - 5$

Step 6

Divide each term in $- p < - 5$ by $- 1$ and simplify.

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Step 6.1

Divide each term in $- p < - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- p}{- 1} > \frac{- 5}{- 1}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Dividing two negative values results in a positive value.

$\frac{p}{1} > \frac{- 5}{- 1}$

Step 6.2.2

Divide $p$ by $1$ .

$p > \frac{- 5}{- 1}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 5$ by $- 1$ .

$p > 5$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$p > 5$

Interval Notation:

$(5, \infty)$