Solve the Inequality for d (d-2)/25>d/10-1/2

$\frac{d - 2}{25} > \frac{d}{10} - \frac{1}{2}$

Step 1

Multiply both sides by $25$ .

$\frac{d - 2}{25} \cdot 25 > (\frac{d}{10} - \frac{1}{2}) \cdot 25$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $25$ .

Cancel the common factor.

$\frac{d - 2}{25} \cdot 25 > (\frac{d}{10} - \frac{1}{2}) \cdot 25$

Rewrite the expression.

$d - 2 > (\frac{d}{10} - \frac{1}{2}) \cdot 25$

Simplify the right side.

Simplify $(\frac{d}{10} - \frac{1}{2}) \cdot 25$ .

Apply the distributive property.

$d - 2 > \frac{d}{10} \cdot 25 - \frac{1}{2} \cdot 25$

Cancel the common factor of $5$ .

Factor $5$ out of $10$ .

$d - 2 > \frac{d}{5 (2)} \cdot 25 - \frac{1}{2} \cdot 25$

Factor $5$ out of $25$ .

$d - 2 > \frac{d}{5 \cdot 2} \cdot (5 \cdot 5) - \frac{1}{2} \cdot 25$

Cancel the common factor.

$d - 2 > \frac{d}{5 \cdot 2} \cdot (5 \cdot 5) - \frac{1}{2} \cdot 25$

Rewrite the expression.

$d - 2 > \frac{d}{2} \cdot 5 - \frac{1}{2} \cdot 25$

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

$d - 2 > \frac{d \cdot 5}{2} - \frac{1}{2} \cdot 25$

Multiply $- \frac{1}{2} \cdot 25$ .

Multiply $25$ by $- 1$ .

$d - 2 > \frac{d \cdot 5}{2} - 25 (\frac{1}{2})$

Combine $- 25$ and $\frac{1}{2}$ .

$d - 2 > \frac{d \cdot 5}{2} + \frac{- 25}{2}$

Simplify each term.

Move $5$ to the left of $d$ .

$d - 2 > \frac{5 \cdot d}{2} + \frac{- 25}{2}$

Move the negative in front of the fraction.

$d - 2 > \frac{5 d}{2} - \frac{25}{2}$

Step 3

Solve for $d$ .

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

Subtract $\frac{5 d}{2}$ from both sides of the inequality.

$d - 2 - \frac{5 d}{2} > - \frac{25}{2}$

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

$d \cdot \frac{2}{2} - \frac{5 d}{2} - 2 > - \frac{25}{2}$

Combine $d$ and $\frac{2}{2}$ .

$\frac{d \cdot 2}{2} - \frac{5 d}{2} - 2 > - \frac{25}{2}$

Combine the numerators over the common denominator.

$\frac{d \cdot 2 - 5 d}{2} - 2 > - \frac{25}{2}$

Subtract $5 d$ from $d \cdot 2$ .

Reorder $d$ and $2$ .

$\frac{2 \cdot d - 5 d}{2} - 2 > - \frac{25}{2}$

Subtract $5 d$ from $2 \cdot d$ .

$\frac{- 3 \cdot d}{2} - 2 > - \frac{25}{2}$

Move the negative in front of the fraction.

$- \frac{3 d}{2} - 2 > - \frac{25}{2}$

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

Add $2$ to both sides of the inequality.

$- \frac{3 d}{2} > - \frac{25}{2} + 2$

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

$- \frac{3 d}{2} > - \frac{25}{2} + 2 \cdot \frac{2}{2}$

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

$- \frac{3 d}{2} > - \frac{25}{2} + \frac{2 \cdot 2}{2}$

Combine the numerators over the common denominator.

$- \frac{3 d}{2} > \frac{- 25 + 2 \cdot 2}{2}$

Simplify the numerator.

Multiply $2$ by $2$ .

$- \frac{3 d}{2} > \frac{- 25 + 4}{2}$

Add $- 25$ and $4$ .

$- \frac{3 d}{2} > \frac{- 21}{2}$

Move the negative in front of the fraction.

$- \frac{3 d}{2} > - \frac{21}{2}$

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

$- 3 d > - 21$

Divide each term in $- 3 d > - 21$ by $- 3$ and simplify.

Divide each term in $- 3 d > - 21$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 d}{- 3} < \frac{- 21}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 d}{- 3} < \frac{- 21}{- 3}$

Divide $d$ by $1$ .

$d < \frac{- 21}{- 3}$

Simplify the right side.

Divide $- 21$ by $- 3$ .

$d < 7$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$d < 7$

Interval Notation:

$(- \infty, 7)$

Solve the Inequality for d (d-2)/25>d/10-1/2

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