Algebra Examples

$\frac{3}{4} (x + 8) > \frac{1}{2} (2 x + 10)$

Step 1

Solve $x < 4$ .

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Step 1.1

Simplify $\frac{3}{4} (x + 8)$ .

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Step 1.1.1

Rewrite.

$0 + 0 + \frac{3}{4} (x + 8) > \frac{1}{2} (2 x + 10)$

Step 1.1.2

Simplify by adding zeros.

$\frac{3}{4} (x + 8) > \frac{1}{2} (2 x + 10)$

Step 1.1.3

Apply the distributive property.

$\frac{3}{4} x + \frac{3}{4} \cdot 8 > \frac{1}{2} (2 x + 10)$

Step 1.1.4

Combine $\frac{3}{4}$ and $x$ .

$\frac{3 x}{4} + \frac{3}{4} \cdot 8 > \frac{1}{2} (2 x + 10)$

Step 1.1.5

Cancel the common factor of $4$ .

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Step 1.1.5.1

Factor $4$ out of $8$ .

$\frac{3 x}{4} + \frac{3}{4} \cdot (4 (2)) > \frac{1}{2} (2 x + 10)$

Step 1.1.5.2

Cancel the common factor.

$\frac{3 x}{4} + \frac{3}{4} \cdot (4 \cdot 2) > \frac{1}{2} (2 x + 10)$

Step 1.1.5.3

Rewrite the expression.

$\frac{3 x}{4} + 3 \cdot 2 > \frac{1}{2} (2 x + 10)$

Step 1.1.6

Multiply $3$ by $2$ .

$\frac{3 x}{4} + 6 > \frac{1}{2} (2 x + 10)$

Step 1.2

Simplify $\frac{1}{2} (2 x + 10)$ .

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Step 1.2.1

Apply the distributive property.

$\frac{3 x}{4} + 6 > \frac{1}{2} (2 x) + \frac{1}{2} \cdot 10$

Step 1.2.2

Cancel the common factor of $2$ .

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Step 1.2.2.1

Factor $2$ out of $2 x$ .

$\frac{3 x}{4} + 6 > \frac{1}{2} (2 (x)) + \frac{1}{2} \cdot 10$

Step 1.2.2.2

Cancel the common factor.

$\frac{3 x}{4} + 6 > \frac{1}{2} (2 x) + \frac{1}{2} \cdot 10$

Step 1.2.2.3

Rewrite the expression.

$\frac{3 x}{4} + 6 > x + \frac{1}{2} \cdot 10$

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Factor $2$ out of $10$ .

$\frac{3 x}{4} + 6 > x + \frac{1}{2} \cdot (2 (5))$

Step 1.2.3.2

Cancel the common factor.

$\frac{3 x}{4} + 6 > x + \frac{1}{2} \cdot (2 \cdot 5)$

Step 1.2.3.3

Rewrite the expression.

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

Step 1.3

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

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Step 1.3.1

Subtract $x$ from both sides of the inequality.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Combine the numerators over the common denominator.

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

Step 1.3.5

Simplify each term.

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Step 1.3.5.1

Simplify the numerator.

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Step 1.3.5.1.1

Factor $x$ out of $3 x - x \cdot 4$ .

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Step 1.3.5.1.1.1

Factor $x$ out of $3 x$ .

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

Step 1.3.5.1.1.2

Factor $x$ out of $- x \cdot 4$ .

$\frac{x \cdot 3 + x (- 1 \cdot 4)}{4} + 6 > 5$

Step 1.3.5.1.1.3

Factor $x$ out of $x \cdot 3 + x (- 1 \cdot 4)$ .

$\frac{x (3 - 1 \cdot 4)}{4} + 6 > 5$

Step 1.3.5.1.2

Multiply $- 1$ by $4$ .

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

Step 1.3.5.1.3

Subtract $4$ from $3$ .

$\frac{x \cdot - 1}{4} + 6 > 5$

Step 1.3.5.2

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

$\frac{- 1 \cdot x}{4} + 6 > 5$

Step 1.3.5.3

Move the negative in front of the fraction.

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

Step 1.4

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

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Step 1.4.1

Subtract $6$ from both sides of the inequality.

$- \frac{x}{4} > 5 - 6$

Step 1.4.2

Subtract $6$ from $5$ .

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

Step 1.5

Divide each term in $- \frac{x}{4} > - 1$ by $- 1$ and simplify.

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Step 1.5.1

Divide each term in $- \frac{x}{4} > - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{4}}{- 1} < \frac{- 1}{- 1}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{4}}{1} < \frac{- 1}{- 1}$

Step 1.5.2.2

Divide $\frac{x}{4}$ by $1$ .

$\frac{x}{4} < \frac{- 1}{- 1}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $- 1$ by $- 1$ .

$\frac{x}{4} < 1$

Step 1.6

Multiply both sides by $4$ .

$\frac{x}{4} \cdot 4 < 1 \cdot 4$

Step 1.7

Simplify.

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Step 1.7.1

Simplify the left side.

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Step 1.7.1.1

Cancel the common factor of $4$ .

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Step 1.7.1.1.1

Cancel the common factor.

$\frac{x}{4} \cdot 4 < 1 \cdot 4$

Step 1.7.1.1.2

Rewrite the expression.

$x < 1 \cdot 4$

Step 1.7.2

Simplify the right side.

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Step 1.7.2.1

Multiply $4$ by $1$ .

$x < 4$

Step 2

Use the inequality $x < 4$ to build the set notation.

${x | x < 4}$

Step 3