Finite Math Examples

$- \frac{2}{3} p - \frac{5}{3} < 2 p$

Step 1

Simplify each term.

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Step 1.1

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

$- \frac{p \cdot 2}{3} - \frac{5}{3} < 2 p$

Step 1.2

Move $2$ to the left of $p$ .

$- \frac{2 p}{3} - \frac{5}{3} < 2 p$

Step 2

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

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Step 2.1

Subtract $2 p$ from both sides of the inequality.

$- \frac{2 p}{3} - \frac{5}{3} - 2 p < 0$

Step 2.2

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

$- \frac{2 p}{3} - 2 p \cdot \frac{3}{3} - \frac{5}{3} < 0$

Step 2.3

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

$- \frac{2 p}{3} + \frac{- 2 p \cdot 3}{3} - \frac{5}{3} < 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{- 2 p - 2 p \cdot 3}{3} - \frac{5}{3} < 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{- 2 p - 2 p \cdot 3 - 5}{3} < 0$

Step 2.6

Multiply $3$ by $- 2$ .

$\frac{- 2 p - 6 p - 5}{3} < 0$

Step 2.7

Subtract $6 p$ from $- 2 p$ .

$\frac{- 8 p - 5}{3} < 0$

Step 2.8

Factor $- 1$ out of $- 8 p$ .

$\frac{- (8 p) - 5}{3} < 0$

Step 2.9

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

$\frac{- (8 p) - 1 (5)}{3} < 0$

Step 2.10

Factor $- 1$ out of $- (8 p) - 1 (5)$ .

$\frac{- (8 p + 5)}{3} < 0$

Step 2.11

Rewrite $- (8 p + 5)$ as $- 1 (8 p + 5)$ .

$\frac{- 1 (8 p + 5)}{3} < 0$

Step 2.12

Move the negative in front of the fraction.

$- \frac{8 p + 5}{3} < 0$

Step 3

Divide each term in $- \frac{8 p + 5}{3} < 0$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \frac{8 p + 5}{3} < 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{8 p + 5}{3}}{- 1} > \frac{0}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{8 p + 5}{3}}{1} > \frac{0}{- 1}$

Step 3.2.2

Divide $\frac{8 p + 5}{3}$ by $1$ .

$\frac{8 p + 5}{3} > \frac{0}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 1$ .

$\frac{8 p + 5}{3} > 0$

Step 4

Multiply both sides by $3$ .

$\frac{8 p + 5}{3} \cdot 3 > 0 \cdot 3$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{8 p + 5}{3} \cdot 3 > 0 \cdot 3$

Step 5.1.1.2

Rewrite the expression.

$8 p + 5 > 0 \cdot 3$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $0$ by $3$ .

$8 p + 5 > 0$

Step 6

Solve for $p$ .

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Step 6.1

Subtract $5$ from both sides of the inequality.

$8 p > - 5$

Step 6.2

Divide each term in $8 p > - 5$ by $8$ and simplify.

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Step 6.2.1

Divide each term in $8 p > - 5$ by $8$ .

$\frac{8 p}{8} > \frac{- 5}{8}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $8$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{8 p}{8} > \frac{- 5}{8}$

Step 6.2.2.1.2

Divide $p$ by $1$ .

$p > \frac{- 5}{8}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Move the negative in front of the fraction.

$p > - \frac{5}{8}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$p > - \frac{5}{8}$

Interval Notation:

$(- \frac{5}{8}, \infty)$