Algebra Examples

$\frac{2 x + 5}{3} + \frac{x}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1

Solve $x < 24$ .

Tap for more steps...

Step 1.1

Simplify $\frac{2 x + 5}{3} + \frac{x}{6}$ .

Tap for more steps...

Step 1.1.1

To write $\frac{2 x + 5}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 x + 5}{3} \cdot \frac{2}{2} + \frac{x}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.2

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.1.2.1

Multiply $\frac{2 x + 5}{3}$ by $\frac{2}{2}$ .

$\frac{(2 x + 5) \cdot 2}{3 \cdot 2} + \frac{x}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.2.2

Multiply $3$ by $2$ .

$\frac{(2 x + 5) \cdot 2}{6} + \frac{x}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.3

Combine the numerators over the common denominator.

$\frac{(2 x + 5) \cdot 2 + x}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.4

Simplify the numerator.

Tap for more steps...

Step 1.1.4.1

Apply the distributive property.

$\frac{2 x \cdot 2 + 5 \cdot 2 + x}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.4.2

Multiply $2$ by $2$ .

$\frac{4 x + 5 \cdot 2 + x}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.4.3

Multiply $5$ by $2$ .

$\frac{4 x + 10 + x}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.4.4

Add $4 x$ and $x$ .

$\frac{5 x + 10}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.4.5

Factor $5$ out of $5 x + 10$ .

Tap for more steps...

Step 1.1.4.5.1

Factor $5$ out of $5 x$ .

$\frac{5 (x) + 10}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.4.5.2

Factor $5$ out of $10$ .

$\frac{5 x + 5 \cdot 2}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.1.4.5.3

Factor $5$ out of $5 x + 5 \cdot 2$ .

$\frac{5 (x + 2)}{6} > \frac{x}{2} + \frac{5 x - 4}{12}$

Step 1.2

Simplify $\frac{x}{2} + \frac{5 x - 4}{12}$ .

Tap for more steps...

Step 1.2.1

To write $\frac{x}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{5 (x + 2)}{6} > \frac{x}{2} \cdot \frac{6}{6} + \frac{5 x - 4}{12}$

Step 1.2.2

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.2.2.1

Multiply $\frac{x}{2}$ by $\frac{6}{6}$ .

$\frac{5 (x + 2)}{6} > \frac{x \cdot 6}{2 \cdot 6} + \frac{5 x - 4}{12}$

Step 1.2.2.2

Multiply $2$ by $6$ .

$\frac{5 (x + 2)}{6} > \frac{x \cdot 6}{12} + \frac{5 x - 4}{12}$

Step 1.2.3

Combine the numerators over the common denominator.

$\frac{5 (x + 2)}{6} > \frac{x \cdot 6 + 5 x - 4}{12}$

Step 1.2.4

Simplify the numerator.

Tap for more steps...

Step 1.2.4.1

Move $6$ to the left of $x$ .

$\frac{5 (x + 2)}{6} > \frac{6 \cdot x + 5 x - 4}{12}$

Step 1.2.4.2

Add $6 x$ and $5 x$ .

$\frac{5 (x + 2)}{6} > \frac{11 x - 4}{12}$

Step 1.3

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

Tap for more steps...

Step 1.3.1

Subtract $\frac{11 x - 4}{12}$ from both sides of the inequality.

$\frac{5 (x + 2)}{6} - \frac{11 x - 4}{12} > 0$

Step 1.3.2

To write $\frac{5 (x + 2)}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{5 (x + 2)}{6} \cdot \frac{2}{2} - \frac{11 x - 4}{12} > 0$

Step 1.3.3

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.3.3.1

Multiply $\frac{5 (x + 2)}{6}$ by $\frac{2}{2}$ .

$\frac{5 (x + 2) \cdot 2}{6 \cdot 2} - \frac{11 x - 4}{12} > 0$

Step 1.3.3.2

Multiply $6$ by $2$ .

$\frac{5 (x + 2) \cdot 2}{12} - \frac{11 x - 4}{12} > 0$

Step 1.3.4

Combine the numerators over the common denominator.

$\frac{5 (x + 2) \cdot 2 - (11 x - 4)}{12} > 0$

Step 1.3.5

Simplify the numerator.

Tap for more steps...

Step 1.3.5.1

Apply the distributive property.

$\frac{(5 x + 5 \cdot 2) \cdot 2 - (11 x - 4)}{12} > 0$

Step 1.3.5.2

Multiply $5$ by $2$ .

$\frac{(5 x + 10) \cdot 2 - (11 x - 4)}{12} > 0$

Step 1.3.5.3

Apply the distributive property.

$\frac{5 x \cdot 2 + 10 \cdot 2 - (11 x - 4)}{12} > 0$

Step 1.3.5.4

Multiply $2$ by $5$ .

$\frac{10 x + 10 \cdot 2 - (11 x - 4)}{12} > 0$

Step 1.3.5.5

Multiply $10$ by $2$ .

$\frac{10 x + 20 - (11 x - 4)}{12} > 0$

Step 1.3.5.6

Apply the distributive property.

$\frac{10 x + 20 - (11 x) - - 4}{12} > 0$

Step 1.3.5.7

Multiply $11$ by $- 1$ .

$\frac{10 x + 20 - 11 x - - 4}{12} > 0$

Step 1.3.5.8

Multiply $- 1$ by $- 4$ .

$\frac{10 x + 20 - 11 x + 4}{12} > 0$

Step 1.3.5.9

Subtract $11 x$ from $10 x$ .

$\frac{- x + 20 + 4}{12} > 0$

Step 1.3.5.10

Add $20$ and $4$ .

$\frac{- x + 24}{12} > 0$

Step 1.3.6

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 24}{12} > 0$

Step 1.3.7

Rewrite $24$ as $- 1 (- 24)$ .

$\frac{- (x) - 1 (- 24)}{12} > 0$

Step 1.3.8

Factor $- 1$ out of $- (x) - 1 (- 24)$ .

$\frac{- (x - 24)}{12} > 0$

Step 1.3.9

Rewrite $- (x - 24)$ as $- 1 (x - 24)$ .

$\frac{- 1 (x - 24)}{12} > 0$

Step 1.3.10

Move the negative in front of the fraction.

$- \frac{x - 24}{12} > 0$

Step 1.4

Divide each term in $- \frac{x - 24}{12} > 0$ by $- 1$ and simplify.

Tap for more steps...

Step 1.4.1

Divide each term in $- \frac{x - 24}{12} > 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x - 24}{12}}{- 1} < \frac{0}{- 1}$

Step 1.4.2

Simplify the left side.

Tap for more steps...

Step 1.4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x - 24}{12}}{1} < \frac{0}{- 1}$

Step 1.4.2.2

Divide $\frac{x - 24}{12}$ by $1$ .

$\frac{x - 24}{12} < \frac{0}{- 1}$

Step 1.4.3

Simplify the right side.

Tap for more steps...

Step 1.4.3.1

Divide $0$ by $- 1$ .

$\frac{x - 24}{12} < 0$

Step 1.5

Multiply both sides by $12$ .

$\frac{x - 24}{12} \cdot 12 < 0 \cdot 12$

Step 1.6

Simplify.

Tap for more steps...

Step 1.6.1

Simplify the left side.

Tap for more steps...

Step 1.6.1.1

Cancel the common factor of $12$ .

Tap for more steps...

Step 1.6.1.1.1

Cancel the common factor.

$\frac{x - 24}{12} \cdot 12 < 0 \cdot 12$

Step 1.6.1.1.2

Rewrite the expression.

$x - 24 < 0 \cdot 12$

Step 1.6.2

Simplify the right side.

Tap for more steps...

Step 1.6.2.1

Multiply $0$ by $12$ .

$x - 24 < 0$

Step 1.7

Add $24$ to both sides of the inequality.

$x < 24$

Step 2

Use the inequality $x < 24$ to build the set notation.

${x | x < 24}$

Step 3