Pre-Algebra Examples

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

Step 1

Simplify $\frac{2 x - 3}{4} + 6$ .

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Step 1.1

To write $6$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 1.2

Simplify terms.

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Step 1.2.1

Combine $6$ and $\frac{4}{4}$ .

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

Step 1.2.2

Combine the numerators over the common denominator.

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

Step 1.3

Simplify the numerator.

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Step 1.3.1

Multiply $6$ by $4$ .

$\frac{2 x - 3 + 24}{4} > 2 + \frac{4 x}{3}$

Step 1.3.2

Add $- 3$ and $24$ .

$\frac{2 x + 21}{4} > 2 + \frac{4 x}{3}$

Step 2

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

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Step 2.1

Subtract $\frac{4 x}{3}$ from both sides of the inequality.

$\frac{2 x + 21}{4} - \frac{4 x}{3} > 2$

Step 2.2

To write $\frac{2 x + 21}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 x + 21}{4} \cdot \frac{3}{3} - \frac{4 x}{3} > 2$

Step 2.3

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

$\frac{2 x + 21}{4} \cdot \frac{3}{3} - \frac{4 x}{3} \cdot \frac{4}{4} > 2$

Step 2.4

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

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Step 2.4.1

Multiply $\frac{2 x + 21}{4}$ by $\frac{3}{3}$ .

$\frac{(2 x + 21) \cdot 3}{4 \cdot 3} - \frac{4 x}{3} \cdot \frac{4}{4} > 2$

Step 2.4.2

Multiply $4$ by $3$ .

$\frac{(2 x + 21) \cdot 3}{12} - \frac{4 x}{3} \cdot \frac{4}{4} > 2$

Step 2.4.3

Multiply $\frac{4 x}{3}$ by $\frac{4}{4}$ .

$\frac{(2 x + 21) \cdot 3}{12} - \frac{4 x \cdot 4}{3 \cdot 4} > 2$

Step 2.4.4

Multiply $3$ by $4$ .

$\frac{(2 x + 21) \cdot 3}{12} - \frac{4 x \cdot 4}{12} > 2$

Step 2.5

Combine the numerators over the common denominator.

$\frac{(2 x + 21) \cdot 3 - 4 x \cdot 4}{12} > 2$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{2 x \cdot 3 + 21 \cdot 3 - 4 x \cdot 4}{12} > 2$

Step 2.6.2

Multiply $3$ by $2$ .

$\frac{6 x + 21 \cdot 3 - 4 x \cdot 4}{12} > 2$

Step 2.6.3

Multiply $21$ by $3$ .

$\frac{6 x + 63 - 4 x \cdot 4}{12} > 2$

Step 2.6.4

Multiply $4$ by $- 4$ .

$\frac{6 x + 63 - 16 x}{12} > 2$

Step 2.6.5

Subtract $16 x$ from $6 x$ .

$\frac{- 10 x + 63}{12} > 2$

Step 2.7

Factor $- 1$ out of $- 10 x$ .

$\frac{- (10 x) + 63}{12} > 2$

Step 2.8

Rewrite $63$ as $- 1 (- 63)$ .

$\frac{- (10 x) - 1 (- 63)}{12} > 2$

Step 2.9

Factor $- 1$ out of $- (10 x) - 1 (- 63)$ .

$\frac{- (10 x - 63)}{12} > 2$

Step 2.10

Rewrite $- (10 x - 63)$ as $- 1 (10 x - 63)$ .

$\frac{- 1 (10 x - 63)}{12} > 2$

Step 2.11

Move the negative in front of the fraction.

$- \frac{10 x - 63}{12} > 2$

Step 3

Divide each term in $- \frac{10 x - 63}{12} > 2$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \frac{10 x - 63}{12} > 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{10 x - 63}{12}}{- 1} < \frac{2}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{10 x - 63}{12}}{1} < \frac{2}{- 1}$

Step 3.2.2

Divide $\frac{10 x - 63}{12}$ by $1$ .

$\frac{10 x - 63}{12} < \frac{2}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $2$ by $- 1$ .

$\frac{10 x - 63}{12} < - 2$

Step 4

Multiply both sides by $12$ .

$\frac{10 x - 63}{12} \cdot 12 < - 2 \cdot 12$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $12$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{10 x - 63}{12} \cdot 12 < - 2 \cdot 12$

Step 5.1.1.2

Rewrite the expression.

$10 x - 63 < - 2 \cdot 12$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 2$ by $12$ .

$10 x - 63 < - 24$

Step 6

Solve for $x$ .

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Step 6.1

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

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Step 6.1.1

Add $63$ to both sides of the inequality.

$10 x < - 24 + 63$

Step 6.1.2

Add $- 24$ and $63$ .

$10 x < 39$

Step 6.2

Divide each term in $10 x < 39$ by $10$ and simplify.

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Step 6.2.1

Divide each term in $10 x < 39$ by $10$ .

$\frac{10 x}{10} < \frac{39}{10}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $10$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} < \frac{39}{10}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{39}{10}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{39}{10}$

Interval Notation:

$(- \infty, \frac{39}{10})$