Solve the Inequality for a (4a-1)/5-(3a-5)/3>a

$\frac{4 a - 1}{5} - \frac{3 a - 5}{3} > a$

Simplify $\frac{4 a - 1}{5} - \frac{3 a - 5}{3}$ .

To write $\frac{4 a - 1}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 a - 1}{5} \cdot \frac{3}{3} - \frac{3 a - 5}{3} > a$

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

$\frac{4 a - 1}{5} \cdot \frac{3}{3} - \frac{3 a - 5}{3} \cdot \frac{5}{5} > a$

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 a - 1}{5}$ by $\frac{3}{3}$ .

$\frac{(4 a - 1) \cdot 3}{5 \cdot 3} - \frac{3 a - 5}{3} \cdot \frac{5}{5} > a$

Multiply $5$ by $3$ .

$\frac{(4 a - 1) \cdot 3}{15} - \frac{3 a - 5}{3} \cdot \frac{5}{5} > a$

Multiply $\frac{3 a - 5}{3}$ by $\frac{5}{5}$ .

$\frac{(4 a - 1) \cdot 3}{15} - \frac{(3 a - 5) \cdot 5}{3 \cdot 5} > a$

Multiply $3$ by $5$ .

$\frac{(4 a - 1) \cdot 3}{15} - \frac{(3 a - 5) \cdot 5}{15} > a$

Combine the numerators over the common denominator.

$\frac{(4 a - 1) \cdot 3 - (3 a - 5) \cdot 5}{15} > a$

Simplify the numerator.

Apply the distributive property.

$\frac{4 a \cdot 3 - 1 \cdot 3 - (3 a - 5) \cdot 5}{15} > a$

Multiply $3$ by $4$ .

$\frac{12 a - 1 \cdot 3 - (3 a - 5) \cdot 5}{15} > a$

Multiply $- 1$ by $3$ .

$\frac{12 a - 3 - (3 a - 5) \cdot 5}{15} > a$

Apply the distributive property.

$\frac{12 a - 3 + (- (3 a) - - 5) \cdot 5}{15} > a$

Multiply $3$ by $- 1$ .

$\frac{12 a - 3 + (- 3 a - - 5) \cdot 5}{15} > a$

Multiply $- 1$ by $- 5$ .

$\frac{12 a - 3 + (- 3 a + 5) \cdot 5}{15} > a$

Apply the distributive property.

$\frac{12 a - 3 - 3 a \cdot 5 + 5 \cdot 5}{15} > a$

Multiply $5$ by $- 3$ .

$\frac{12 a - 3 - 15 a + 5 \cdot 5}{15} > a$

Multiply $5$ by $5$ .

$\frac{12 a - 3 - 15 a + 25}{15} > a$

Subtract $15 a$ from $12 a$ .

$\frac{- 3 a - 3 + 25}{15} > a$

Add $- 3$ and $25$ .

$\frac{- 3 a + 22}{15} > a$

Simplify with factoring out.

Factor $- 1$ out of $- 3 a$ .

$\frac{- (3 a) + 22}{15} > a$

Rewrite $22$ as $- 1 (- 22)$ .

$\frac{- (3 a) - 1 (- 22)}{15} > a$

Factor $- 1$ out of $- (3 a) - 1 (- 22)$ .

$\frac{- (3 a - 22)}{15} > a$

Simplify the expression.

Rewrite $- (3 a - 22)$ as $- 1 (3 a - 22)$ .

$\frac{- 1 (3 a - 22)}{15} > a$

Move the negative in front of the fraction.

$- \frac{3 a - 22}{15} > a$

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

Subtract $a$ from both sides of the inequality.

$- \frac{3 a - 22}{15} - a > 0$

To write $- a$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$- \frac{3 a - 22}{15} - a \cdot \frac{15}{15} > 0$

Combine $- a$ and $\frac{15}{15}$ .

$- \frac{3 a - 22}{15} + \frac{- a \cdot 15}{15} > 0$

Combine the numerators over the common denominator.

$\frac{- (3 a - 22) - a \cdot 15}{15} > 0$

Simplify the numerator.

Apply the distributive property.

$\frac{- (3 a) - - 22 - a \cdot 15}{15} > 0$

Multiply $3$ by $- 1$ .

$\frac{- 3 a - - 22 - a \cdot 15}{15} > 0$

Multiply $- 1$ by $- 22$ .

$\frac{- 3 a + 22 - a \cdot 15}{15} > 0$

Multiply $15$ by $- 1$ .

$\frac{- 3 a + 22 - 15 a}{15} > 0$

Subtract $15 a$ from $- 3 a$ .

$\frac{- 18 a + 22}{15} > 0$

Factor $2$ out of $- 18 a + 22$ .

Factor $2$ out of $- 18 a$ .

$\frac{2 (- 9 a) + 22}{15} > 0$

Factor $2$ out of $22$ .

$\frac{2 (- 9 a) + 2 (11)}{15} > 0$

Factor $2$ out of $2 (- 9 a) + 2 (11)$ .

$\frac{2 (- 9 a + 11)}{15} > 0$

Factor $- 1$ out of $- 9 a$ .

$\frac{2 (- (9 a) + 11)}{15} > 0$

Rewrite $11$ as $- 1 (- 11)$ .

$\frac{2 (- (9 a) - 1 (- 11))}{15} > 0$

Factor $- 1$ out of $- (9 a) - 1 (- 11)$ .

$\frac{2 (- (9 a - 11))}{15} > 0$

Rewrite $- (9 a - 11)$ as $- 1 (9 a - 11)$ .

$\frac{2 (- 1 (9 a - 11))}{15} > 0$

Move the negative in front of the fraction.

$- \frac{2 (9 a - 11)}{15} > 0$

Divide each term in $- \frac{2 (9 a - 11)}{15} > 0$ by $- 1$ and simplify.

Divide each term in $- \frac{2 (9 a - 11)}{15} > 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 (9 a - 11)}{15}}{- 1} < \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{\frac{2 (9 a - 11)}{15}}{1} < \frac{0}{- 1}$

Divide $\frac{2 (9 a - 11)}{15}$ by $1$ .

$\frac{2 (9 a - 11)}{15} < \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$\frac{2 (9 a - 11)}{15} < 0$

Multiply both sides by $15$ .

$\frac{2 (9 a - 11)}{15} \cdot 15 < 0 \cdot 15$

Simplify.

Simplify the left side.

Simplify $\frac{2 (9 a - 11)}{15} \cdot 15$ .

Cancel the common factor of $15$ .

Cancel the common factor.

$\frac{2 (9 a - 11)}{15} \cdot 15 < 0 \cdot 15$

Rewrite the expression.

$2 (9 a - 11) < 0 \cdot 15$

Apply the distributive property.

$2 (9 a) + 2 \cdot - 11 < 0 \cdot 15$

Multiply.

Multiply $9$ by $2$ .

$18 a + 2 \cdot - 11 < 0 \cdot 15$

Multiply $2$ by $- 11$ .

$18 a - 22 < 0 \cdot 15$

Simplify the right side.

Multiply $0$ by $15$ .

$18 a - 22 < 0$

Solve for $a$ .

Add $22$ to both sides of the inequality.

$18 a < 22$

Divide each term in $18 a < 22$ by $18$ and simplify.

Divide each term in $18 a < 22$ by $18$ .

$\frac{18 a}{18} < \frac{22}{18}$

Simplify the left side.

Cancel the common factor of $18$ .

Cancel the common factor.

$\frac{18 a}{18} < \frac{22}{18}$

Divide $a$ by $1$ .

$a < \frac{22}{18}$

Simplify the right side.

Cancel the common factor of $22$ and $18$ .

Factor $2$ out of $22$ .

$a < \frac{2 (11)}{18}$

Cancel the common factors.

Factor $2$ out of $18$ .

$a < \frac{2 \cdot 11}{2 \cdot 9}$

Cancel the common factor.

$a < \frac{2 \cdot 11}{2 \cdot 9}$

Rewrite the expression.

$a < \frac{11}{9}$

The result can be shown in multiple forms.

Inequality Form:

$a < \frac{11}{9}$

Interval Notation:

$(- \infty, \frac{11}{9})$

Solve the Inequality for a (4a-1)/5-(3a-5)/3>a

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