Precalculus Examples

$\frac{x}{3} - \frac{3}{4} - 1 > \frac{x}{2}$

Step 1

Multiply both sides by $2$ .

$(\frac{x}{3} - \frac{3}{4} - 1) \cdot 2 > \frac{x}{2} \cdot 2$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Simplify the left side.

Tap for more steps...

Step 2.1.1

Simplify $(\frac{x}{3} - \frac{3}{4} - 1) \cdot 2$ .

Tap for more steps...

Step 2.1.1.1

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

$(\frac{x}{3} - \frac{3}{4} - 1 \cdot \frac{4}{4}) \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.2

Combine $- 1$ and $\frac{4}{4}$ .

$(\frac{x}{3} - \frac{3}{4} + \frac{- 1 \cdot 4}{4}) \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.3

Combine the numerators over the common denominator.

$(\frac{x}{3} + \frac{- 3 - 1 \cdot 4}{4}) \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.4

Simplify the numerator.

Tap for more steps...

Step 2.1.1.4.1

Multiply $- 1$ by $4$ .

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

Step 2.1.1.4.2

Subtract $4$ from $- 3$ .

$(\frac{x}{3} + \frac{- 7}{4}) \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.5

Simplify terms.

Tap for more steps...

Step 2.1.1.5.1

Move the negative in front of the fraction.

$(\frac{x}{3} - \frac{7}{4}) \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.5.2

Apply the distributive property.

$\frac{x}{3} \cdot 2 - \frac{7}{4} \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.5.3

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

$\frac{x \cdot 2}{3} - \frac{7}{4} \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.1.5.4.1

Move the leading negative in $- \frac{7}{4}$ into the numerator.

$\frac{x \cdot 2}{3} + \frac{- 7}{4} \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.5.4.2

Factor $2$ out of $4$ .

$\frac{x \cdot 2}{3} + \frac{- 7}{2 (2)} \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.5.4.3

Cancel the common factor.

$\frac{x \cdot 2}{3} + \frac{- 7}{2 \cdot 2} \cdot 2 > \frac{x}{2} \cdot 2$

Step 2.1.1.5.4.4

Rewrite the expression.

$\frac{x \cdot 2}{3} + \frac{- 7}{2} > \frac{x}{2} \cdot 2$

Step 2.1.1.6

Simplify each term.

Tap for more steps...

Step 2.1.1.6.1

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

$\frac{2 \cdot x}{3} + \frac{- 7}{2} > \frac{x}{2} \cdot 2$

Step 2.1.1.6.2

Move the negative in front of the fraction.

$\frac{2 x}{3} - \frac{7}{2} > \frac{x}{2} \cdot 2$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{2 x}{3} - \frac{7}{2} > \frac{x}{2} \cdot 2$

Step 2.2.1.2

Rewrite the expression.

$\frac{2 x}{3} - \frac{7}{2} > x$

Step 3

Solve for $x$ .

Tap for more steps...

Step 3.1

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

Tap for more steps...

Step 3.1.1

Subtract $x$ from both sides of the inequality.

$\frac{2 x}{3} - \frac{7}{2} - x > 0$

Step 3.1.2

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

$\frac{2 x}{3} - x \cdot \frac{3}{3} - \frac{7}{2} > 0$

Step 3.1.3

Combine $- x$ and $\frac{3}{3}$ .

$\frac{2 x}{3} + \frac{- x \cdot 3}{3} - \frac{7}{2} > 0$

Step 3.1.4

Combine the numerators over the common denominator.

$\frac{2 x - x \cdot 3}{3} - \frac{7}{2} > 0$

Step 3.1.5

Simplify each term.

Tap for more steps...

Step 3.1.5.1

Simplify the numerator.

Tap for more steps...

Step 3.1.5.1.1

Factor $x$ out of $2 x - x \cdot 3$ .

Tap for more steps...

Step 3.1.5.1.1.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2 - x \cdot 3}{3} - \frac{7}{2} > 0$

Step 3.1.5.1.1.2

Factor $x$ out of $- x \cdot 3$ .

$\frac{x \cdot 2 + x (- 1 \cdot 3)}{3} - \frac{7}{2} > 0$

Step 3.1.5.1.1.3

Factor $x$ out of $x \cdot 2 + x (- 1 \cdot 3)$ .

$\frac{x (2 - 1 \cdot 3)}{3} - \frac{7}{2} > 0$

Step 3.1.5.1.2

Multiply $- 1$ by $3$ .

$\frac{x (2 - 3)}{3} - \frac{7}{2} > 0$

Step 3.1.5.1.3

Subtract $3$ from $2$ .

$\frac{x \cdot - 1}{3} - \frac{7}{2} > 0$

Step 3.1.5.2

Move $- 1$ to the left of $x$ .

$\frac{- 1 \cdot x}{3} - \frac{7}{2} > 0$

Step 3.1.5.3

Move the negative in front of the fraction.

$- \frac{x}{3} - \frac{7}{2} > 0$

Step 3.2

Add $\frac{7}{2}$ to both sides of the inequality.

$- \frac{x}{3} > \frac{7}{2}$

Step 3.3

Divide each term in $- \frac{x}{3} > \frac{7}{2}$ by $- 1$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $- \frac{x}{3} > \frac{7}{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{3}}{- 1} < \frac{\frac{7}{2}}{- 1}$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{3}}{1} < \frac{\frac{7}{2}}{- 1}$

Step 3.3.2.2

Divide $\frac{x}{3}$ by $1$ .

$\frac{x}{3} < \frac{\frac{7}{2}}{- 1}$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Move the negative one from the denominator of $\frac{\frac{7}{2}}{- 1}$ .

$\frac{x}{3} < - 1 \cdot \frac{7}{2}$

Step 3.3.3.2

Rewrite $- 1 \cdot \frac{7}{2}$ as $- \frac{7}{2}$ .

$\frac{x}{3} < - \frac{7}{2}$

Step 3.4

Multiply both sides by $3$ .

$\frac{x}{3} \cdot 3 < - \frac{7}{2} \cdot 3$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Simplify the left side.

Tap for more steps...

Step 3.5.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.5.1.1.1

Cancel the common factor.

$\frac{x}{3} \cdot 3 < - \frac{7}{2} \cdot 3$

Step 3.5.1.1.2

Rewrite the expression.

$x < - \frac{7}{2} \cdot 3$

Step 3.5.2

Simplify the right side.

Tap for more steps...

Step 3.5.2.1

Simplify $- \frac{7}{2} \cdot 3$ .

Tap for more steps...

Step 3.5.2.1.1

Multiply $- \frac{7}{2} \cdot 3$ .

Tap for more steps...

Step 3.5.2.1.1.1

Multiply $3$ by $- 1$ .

$x < - 3 (\frac{7}{2})$

Step 3.5.2.1.1.2

Combine $- 3$ and $\frac{7}{2}$ .

$x < \frac{- 3 \cdot 7}{2}$

Step 3.5.2.1.1.3

Multiply $- 3$ by $7$ .

$x < \frac{- 21}{2}$

Step 3.5.2.1.2

Move the negative in front of the fraction.

$x < - \frac{21}{2}$

Step 4

Convert the inequality to interval notation.

$(- \infty, - \frac{21}{2})$

Step 5