Algebra Examples

$- \frac{6}{5} (4 - x) \leq - 4 (x + \frac{1}{2})$

Step 1

Simplify $- \frac{6}{5} \cdot (4 - x)$ .

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Step 1.1

Rewrite.

$0 + 0 - \frac{6}{5} \cdot (4 - x) \leq - 4 (x + \frac{1}{2})$

Step 1.2

Simplify by adding zeros.

$- \frac{6}{5} \cdot (4 - x) \leq - 4 (x + \frac{1}{2})$

Step 1.3

Apply the distributive property.

$- \frac{6}{5} \cdot 4 - \frac{6}{5} (- x) \leq - 4 (x + \frac{1}{2})$

Step 1.4

Multiply $- \frac{6}{5} \cdot 4$ .

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Step 1.4.1

Multiply $4$ by $- 1$ .

$- 4 (\frac{6}{5}) - \frac{6}{5} (- x) \leq - 4 (x + \frac{1}{2})$

Step 1.4.2

Combine $- 4$ and $\frac{6}{5}$ .

$\frac{- 4 \cdot 6}{5} - \frac{6}{5} (- x) \leq - 4 (x + \frac{1}{2})$

Step 1.4.3

Multiply $- 4$ by $6$ .

$\frac{- 24}{5} - \frac{6}{5} (- x) \leq - 4 (x + \frac{1}{2})$

Step 1.5

Multiply $- \frac{6}{5} (- x)$ .

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Step 1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{- 24}{5} + 1 (\frac{6}{5}) x \leq - 4 (x + \frac{1}{2})$

Step 1.5.2

Multiply $\frac{6}{5}$ by $1$ .

$\frac{- 24}{5} + \frac{6}{5} x \leq - 4 (x + \frac{1}{2})$

Step 1.5.3

Combine $\frac{6}{5}$ and $x$ .

$\frac{- 24}{5} + \frac{6 x}{5} \leq - 4 (x + \frac{1}{2})$

Step 1.6

Move the negative in front of the fraction.

$- \frac{24}{5} + \frac{6 x}{5} \leq - 4 (x + \frac{1}{2})$

Step 2

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

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Step 2.1

Apply the distributive property.

$- \frac{24}{5} + \frac{6 x}{5} \leq - 4 x - 4 (\frac{1}{2})$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $- 4$ .

$- \frac{24}{5} + \frac{6 x}{5} \leq - 4 x + 2 (- 2) \frac{1}{2}$

Step 2.2.2

Cancel the common factor.

$- \frac{24}{5} + \frac{6 x}{5} \leq - 4 x + 2 \cdot - 2 \frac{1}{2}$

Step 2.2.3

Rewrite the expression.

$- \frac{24}{5} + \frac{6 x}{5} \leq - 4 x - 2$

Step 3

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

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Step 3.1

Add $4 x$ to both sides of the inequality.

$- \frac{24}{5} + \frac{6 x}{5} + 4 x \leq - 2$

Step 3.2

To write $4 x$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{24}{5} + \frac{6 x}{5} + 4 x \cdot \frac{5}{5} \leq - 2$

Step 3.3

Combine $4 x$ and $\frac{5}{5}$ .

$- \frac{24}{5} + \frac{6 x}{5} + \frac{4 x \cdot 5}{5} \leq - 2$

Step 3.4

Combine the numerators over the common denominator.

$- \frac{24}{5} + \frac{6 x + 4 x \cdot 5}{5} \leq - 2$

Step 3.5

Combine the numerators over the common denominator.

$\frac{- 24 + 6 x + 4 x \cdot 5}{5} \leq - 2$

Step 3.6

Multiply $5$ by $4$ .

$\frac{- 24 + 6 x + 20 x}{5} \leq - 2$

Step 3.7

Add $6 x$ and $20 x$ .

$\frac{- 24 + 26 x}{5} \leq - 2$

Step 3.8

Factor $2$ out of $- 24 + 26 x$ .

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Step 3.8.1

Factor $2$ out of $- 24$ .

$\frac{2 (- 12) + 26 x}{5} \leq - 2$

Step 3.8.2

Factor $2$ out of $26 x$ .

$\frac{2 (- 12) + 2 (13 x)}{5} \leq - 2$

Step 3.8.3

Factor $2$ out of $2 (- 12) + 2 (13 x)$ .

$\frac{2 (- 12 + 13 x)}{5} \leq - 2$

Step 3.9

Rewrite $- 12$ as $- 1 (12)$ .

$\frac{2 (- 1 (12) + 13 x)}{5} \leq - 2$

Step 3.10

Factor $- 1$ out of $13 x$ .

$\frac{2 (- 1 (12) - (- 13 x))}{5} \leq - 2$

Step 3.11

Factor $- 1$ out of $- 1 (12) - (- 13 x)$ .

$\frac{2 (- 1 (12 - 13 x))}{5} \leq - 2$

Step 3.12

Move the negative in front of the fraction.

$- \frac{2 (12 - 13 x)}{5} \leq - 2$

Step 4

Divide each term in $- \frac{2 (12 - 13 x)}{5} \leq - 2$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{2 (12 - 13 x)}{5} \leq - 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{2 (12 - 13 x)}{5}}{- 1} \geq \frac{- 2}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{2 (12 - 13 x)}{5}}{1} \geq \frac{- 2}{- 1}$

Step 4.2.2

Divide $\frac{2 (12 - 13 x)}{5}$ by $1$ .

$\frac{2 (12 - 13 x)}{5} \geq \frac{- 2}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $- 2$ by $- 1$ .

$\frac{2 (12 - 13 x)}{5} \geq 2$

Step 5

Multiply both sides by $5$ .

$\frac{2 (12 - 13 x)}{5} \cdot 5 \geq 2 \cdot 5$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{2 (12 - 13 x)}{5} \cdot 5$ .

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Step 6.1.1.1

Cancel the common factor of $5$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{2 (12 - 13 x)}{5} \cdot 5 \geq 2 \cdot 5$

Step 6.1.1.1.2

Rewrite the expression.

$2 (12 - 13 x) \geq 2 \cdot 5$

Step 6.1.1.2

Apply the distributive property.

$2 \cdot 12 + 2 (- 13 x) \geq 2 \cdot 5$

Step 6.1.1.3

Simplify the expression.

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Step 6.1.1.3.1

Multiply $2$ by $12$ .

$24 + 2 (- 13 x) \geq 2 \cdot 5$

Step 6.1.1.3.2

Multiply $- 13$ by $2$ .

$24 - 26 x \geq 2 \cdot 5$

Step 6.1.1.3.3

Reorder $24$ and $- 26 x$ .

$- 26 x + 24 \geq 2 \cdot 5$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $2$ by $5$ .

$- 26 x + 24 \geq 10$

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

Subtract $24$ from both sides of the inequality.

$- 26 x \geq 10 - 24$

Step 7.1.2

Subtract $24$ from $10$ .

$- 26 x \geq - 14$

Step 7.2

Divide each term in $- 26 x \geq - 14$ by $- 26$ and simplify.

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Step 7.2.1

Divide each term in $- 26 x \geq - 14$ by $- 26$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 26 x}{- 26} \leq \frac{- 14}{- 26}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $- 26$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{- 26 x}{- 26} \leq \frac{- 14}{- 26}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 14}{- 26}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Cancel the common factor of $- 14$ and $- 26$ .

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Step 7.2.3.1.1

Factor $- 2$ out of $- 14$ .

$x \leq \frac{- 2 (7)}{- 26}$

Step 7.2.3.1.2

Cancel the common factors.

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Step 7.2.3.1.2.1

Factor $- 2$ out of $- 26$ .

$x \leq \frac{- 2 \cdot 7}{- 2 \cdot 13}$

Step 7.2.3.1.2.2

Cancel the common factor.

$x \leq \frac{- 2 \cdot 7}{- 2 \cdot 13}$

Step 7.2.3.1.2.3

Rewrite the expression.

$x \leq \frac{7}{13}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x \leq \frac{7}{13}$

Interval Notation:

$(- \infty, \frac{7}{13}]$