Precalculus Examples

$\frac{2 x}{4} - \frac{5 x + 1}{3} > 3$

Step 1

Simplify $\frac{2 x}{4} - \frac{5 x + 1}{3}$ .

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Step 1.1

Cancel the common factor of $2$ and $4$ .

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Step 1.1.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{4} - \frac{5 x + 1}{3} > 3$

Step 1.1.2

Cancel the common factors.

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Step 1.1.2.1

Factor $2$ out of $4$ .

$\frac{2 x}{2 \cdot 2} - \frac{5 x + 1}{3} > 3$

Step 1.1.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 2} - \frac{5 x + 1}{3} > 3$

Step 1.1.2.3

Rewrite the expression.

$\frac{x}{2} - \frac{5 x + 1}{3} > 3$

Step 1.2

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

$\frac{x}{2} \cdot \frac{3}{3} - \frac{5 x + 1}{3} > 3$

Step 1.3

To write $- \frac{5 x + 1}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x}{2} \cdot \frac{3}{3} - \frac{5 x + 1}{3} \cdot \frac{2}{2} > 3$

Step 1.4

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

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Step 1.4.1

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

$\frac{x \cdot 3}{2 \cdot 3} - \frac{5 x + 1}{3} \cdot \frac{2}{2} > 3$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{x \cdot 3}{6} - \frac{5 x + 1}{3} \cdot \frac{2}{2} > 3$

Step 1.4.3

Multiply $\frac{5 x + 1}{3}$ by $\frac{2}{2}$ .

$\frac{x \cdot 3}{6} - \frac{(5 x + 1) \cdot 2}{3 \cdot 2} > 3$

Step 1.4.4

Multiply $3$ by $2$ .

$\frac{x \cdot 3}{6} - \frac{(5 x + 1) \cdot 2}{6} > 3$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x \cdot 3 - (5 x + 1) \cdot 2}{6} > 3$

Step 1.6

Simplify the numerator.

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Step 1.6.1

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

$\frac{3 \cdot x - (5 x + 1) \cdot 2}{6} > 3$

Step 1.6.2

Apply the distributive property.

$\frac{3 x + (- (5 x) - 1 \cdot 1) \cdot 2}{6} > 3$

Step 1.6.3

Multiply $5$ by $- 1$ .

$\frac{3 x + (- 5 x - 1 \cdot 1) \cdot 2}{6} > 3$

Step 1.6.4

Multiply $- 1$ by $1$ .

$\frac{3 x + (- 5 x - 1) \cdot 2}{6} > 3$

Step 1.6.5

Apply the distributive property.

$\frac{3 x - 5 x \cdot 2 - 1 \cdot 2}{6} > 3$

Step 1.6.6

Multiply $2$ by $- 5$ .

$\frac{3 x - 10 x - 1 \cdot 2}{6} > 3$

Step 1.6.7

Multiply $- 1$ by $2$ .

$\frac{3 x - 10 x - 2}{6} > 3$

Step 1.6.8

Subtract $10 x$ from $3 x$ .

$\frac{- 7 x - 2}{6} > 3$

Step 1.7

Simplify with factoring out.

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Step 1.7.1

Factor $- 1$ out of $- 7 x$ .

$\frac{- (7 x) - 2}{6} > 3$

Step 1.7.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- (7 x) - 1 (2)}{6} > 3$

Step 1.7.3

Factor $- 1$ out of $- (7 x) - 1 (2)$ .

$\frac{- (7 x + 2)}{6} > 3$

Step 1.7.4

Simplify the expression.

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Step 1.7.4.1

Rewrite $- (7 x + 2)$ as $- 1 (7 x + 2)$ .

$\frac{- 1 (7 x + 2)}{6} > 3$

Step 1.7.4.2

Move the negative in front of the fraction.

$- \frac{7 x + 2}{6} > 3$

Step 2

Divide each term in $- \frac{7 x + 2}{6} > 3$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- \frac{7 x + 2}{6} > 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{7 x + 2}{6}}{- 1} < \frac{3}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{7 x + 2}{6}}{1} < \frac{3}{- 1}$

Step 2.2.2

Divide $\frac{7 x + 2}{6}$ by $1$ .

$\frac{7 x + 2}{6} < \frac{3}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $3$ by $- 1$ .

$\frac{7 x + 2}{6} < - 3$

Step 3

Multiply both sides by $6$ .

$\frac{7 x + 2}{6} \cdot 6 < - 3 \cdot 6$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $6$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{7 x + 2}{6} \cdot 6 < - 3 \cdot 6$

Step 4.1.1.2

Rewrite the expression.

$7 x + 2 < - 3 \cdot 6$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 3$ by $6$ .

$7 x + 2 < - 18$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

Subtract $2$ from both sides of the inequality.

$7 x < - 18 - 2$

Step 5.1.2

Subtract $2$ from $- 18$ .

$7 x < - 20$

Step 5.2

Divide each term in $7 x < - 20$ by $7$ and simplify.

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Step 5.2.1

Divide each term in $7 x < - 20$ by $7$ .

$\frac{7 x}{7} < \frac{- 20}{7}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $7$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{- 20}{7}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 20}{7}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$x < - \frac{20}{7}$

Step 6

Convert the inequality to interval notation.

$(- \infty, - \frac{20}{7})$

Step 7