Solve for a (a+2)/16-(a+3)/12>(a+6)/24

$\frac{a + 2}{16} - \frac{a + 3}{12} > \frac{a + 6}{24}$

Multiply both sides by $24$ .

$(\frac{a + 2}{16} - \frac{a + 3}{12}) \cdot 24 > \frac{a + 6}{24} \cdot 24$

Simplify.

Simplify the left side.

Simplify $(\frac{a + 2}{16} - \frac{a + 3}{12}) \cdot 24$ .

To write $\frac{a + 2}{16}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(\frac{a + 2}{16} \cdot \frac{3}{3} - \frac{a + 3}{12}) \cdot 24 > \frac{a + 6}{24} \cdot 24$

To write $- \frac{a + 3}{12}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(\frac{a + 2}{16} \cdot \frac{3}{3} - \frac{a + 3}{12} \cdot \frac{4}{4}) \cdot 24 > \frac{a + 6}{24} \cdot 24$

Write each expression with a common denominator of $48$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a + 2}{16}$ by $\frac{3}{3}$ .

$(\frac{(a + 2) \cdot 3}{16 \cdot 3} - \frac{a + 3}{12} \cdot \frac{4}{4}) \cdot 24 > \frac{a + 6}{24} \cdot 24$

Multiply $16$ by $3$ .

$(\frac{(a + 2) \cdot 3}{48} - \frac{a + 3}{12} \cdot \frac{4}{4}) \cdot 24 > \frac{a + 6}{24} \cdot 24$

Multiply $\frac{a + 3}{12}$ by $\frac{4}{4}$ .

$(\frac{(a + 2) \cdot 3}{48} - \frac{(a + 3) \cdot 4}{12 \cdot 4}) \cdot 24 > \frac{a + 6}{24} \cdot 24$

Multiply $12$ by $4$ .

$(\frac{(a + 2) \cdot 3}{48} - \frac{(a + 3) \cdot 4}{48}) \cdot 24 > \frac{a + 6}{24} \cdot 24$

Combine the numerators over the common denominator.

$\frac{(a + 2) \cdot 3 - (a + 3) \cdot 4}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Simplify the numerator.

Apply the distributive property.

$\frac{a \cdot 3 + 2 \cdot 3 - (a + 3) \cdot 4}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

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

$\frac{3 \cdot a + 2 \cdot 3 - (a + 3) \cdot 4}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Multiply $2$ by $3$ .

$\frac{3 \cdot a + 6 - (a + 3) \cdot 4}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Apply the distributive property.

$\frac{3 a + 6 + (- a - 1 \cdot 3) \cdot 4}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Multiply $- 1$ by $3$ .

$\frac{3 a + 6 + (- a - 3) \cdot 4}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Apply the distributive property.

$\frac{3 a + 6 - a \cdot 4 - 3 \cdot 4}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Multiply $4$ by $- 1$ .

$\frac{3 a + 6 - 4 a - 3 \cdot 4}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Multiply $- 3$ by $4$ .

$\frac{3 a + 6 - 4 a - 12}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Subtract $4 a$ from $3 a$ .

$\frac{- a + 6 - 12}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Subtract $12$ from $6$ .

$\frac{- a - 6}{48} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Simplify terms.

Cancel the common factor of $24$ .

Factor $24$ out of $48$ .

$\frac{- a - 6}{24 (2)} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Cancel the common factor.

$\frac{- a - 6}{24 \cdot 2} \cdot 24 > \frac{a + 6}{24} \cdot 24$

Rewrite the expression.

$\frac{- a - 6}{2} > \frac{a + 6}{24} \cdot 24$

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

$\frac{- (a) - 6}{2} > \frac{a + 6}{24} \cdot 24$

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- (a) - 1 (6)}{2} > \frac{a + 6}{24} \cdot 24$

Factor $- 1$ out of $- (a) - 1 (6)$ .

$\frac{- (a + 6)}{2} > \frac{a + 6}{24} \cdot 24$

Simplify the expression.

Rewrite $- (a + 6)$ as $- 1 (a + 6)$ .

$\frac{- 1 (a + 6)}{2} > \frac{a + 6}{24} \cdot 24$

Move the negative in front of the fraction.

$- \frac{a + 6}{2} > \frac{a + 6}{24} \cdot 24$

Simplify the right side.

Cancel the common factor of $24$ .

Cancel the common factor.

$- \frac{a + 6}{2} > \frac{a + 6}{24} \cdot 24$

Rewrite the expression.

$- \frac{a + 6}{2} > a + 6$

Solve for $a$ .

Multiply both sides by $2$ .

$- \frac{a + 6}{2} \cdot 2 > (a + 6) \cdot 2$

Simplify.

Simplify the left side.

Simplify $- \frac{a + 6}{2} \cdot 2$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{a + 6}{2}$ into the numerator.

$\frac{- (a + 6)}{2} \cdot 2 > (a + 6) \cdot 2$

Cancel the common factor.

$\frac{- (a + 6)}{2} \cdot 2 > (a + 6) \cdot 2$

Rewrite the expression.

$- (a + 6) > (a + 6) \cdot 2$

Apply the distributive property.

$- a - 1 \cdot 6 > (a + 6) \cdot 2$

Multiply $- 1$ by $6$ .

$- a - 6 > (a + 6) \cdot 2$

Simplify the right side.

Simplify $(a + 6) \cdot 2$ .

Apply the distributive property.

$- a - 6 > a \cdot 2 + 6 \cdot 2$

Simplify the expression.

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

$- a - 6 > 2 \cdot a + 6 \cdot 2$

Multiply $6$ by $2$ .

$- a - 6 > 2 a + 12$

Solve for $a$ .

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

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

$- a - 6 - 2 a > 12$

Subtract $2 a$ from $- a$ .

$- 3 a - 6 > 12$

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

Add $6$ to both sides of the inequality.

$- 3 a > 12 + 6$

Add $12$ and $6$ .

$- 3 a > 18$

Divide each term in $- 3 a > 18$ by $- 3$ and simplify.

Divide each term in $- 3 a > 18$ by $- 3$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 3 a}{- 3} < \frac{18}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 a}{- 3} < \frac{18}{- 3}$

Divide $a$ by $1$ .

$a < \frac{18}{- 3}$

Simplify the right side.

Divide $18$ by $- 3$ .

$a < - 6$

The result can be shown in multiple forms.

Inequality Form:

$a < - 6$

Interval Notation:

$(- \infty, - 6)$

Solve for a (a+2)/16-(a+3)/12>(a+6)/24

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