Algebra Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify each term.

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Step 1.1.1.1

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

$- 2 (- \frac{x \cdot 4}{5} + 3) - x \leq 3 x$

Step 1.1.1.2

Move $4$ to the left of $x$ .

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

Step 1.1.2

Apply the distributive property.

$- 2 (- \frac{4 x}{5}) - 2 \cdot 3 - x \leq 3 x$

Step 1.1.3

Multiply $- 2 (- \frac{4 x}{5})$ .

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Step 1.1.3.1

Multiply $- 1$ by $- 2$ .

$2 \frac{4 x}{5} - 2 \cdot 3 - x \leq 3 x$

Step 1.1.3.2

Combine $2$ and $\frac{4 x}{5}$ .

$\frac{2 (4 x)}{5} - 2 \cdot 3 - x \leq 3 x$

Step 1.1.3.3

Multiply $4$ by $2$ .

$\frac{8 x}{5} - 2 \cdot 3 - x \leq 3 x$

Step 1.1.4

Multiply $- 2$ by $3$ .

$\frac{8 x}{5} - 6 - x \leq 3 x$

Step 1.2

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

$\frac{8 x}{5} - x \cdot \frac{5}{5} - 6 \leq 3 x$

Step 1.3

Simplify terms.

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Step 1.3.1

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

$\frac{8 x}{5} + \frac{- x \cdot 5}{5} - 6 \leq 3 x$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{8 x - x \cdot 5}{5} - 6 \leq 3 x$

Step 1.4

Simplify each term.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Factor $x$ out of $8 x - x \cdot 5$ .

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Step 1.4.1.1.1

Factor $x$ out of $8 x$ .

$\frac{x \cdot 8 - x \cdot 5}{5} - 6 \leq 3 x$

Step 1.4.1.1.2

Factor $x$ out of $- x \cdot 5$ .

$\frac{x \cdot 8 + x (- 1 \cdot 5)}{5} - 6 \leq 3 x$

Step 1.4.1.1.3

Factor $x$ out of $x \cdot 8 + x (- 1 \cdot 5)$ .

$\frac{x (8 - 1 \cdot 5)}{5} - 6 \leq 3 x$

Step 1.4.1.2

Multiply $- 1$ by $5$ .

$\frac{x (8 - 5)}{5} - 6 \leq 3 x$

Step 1.4.1.3

Subtract $5$ from $8$ .

$\frac{x \cdot 3}{5} - 6 \leq 3 x$

Step 1.4.2

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

$\frac{3 x}{5} - 6 \leq 3 x$

Step 2

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

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Step 2.1

Subtract $3 x$ from both sides of the inequality.

$\frac{3 x}{5} - 6 - 3 x \leq 0$

Step 2.2

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

$\frac{3 x}{5} - 3 x \cdot \frac{5}{5} - 6 \leq 0$

Step 2.3

Combine $- 3 x$ and $\frac{5}{5}$ .

$\frac{3 x}{5} + \frac{- 3 x \cdot 5}{5} - 6 \leq 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{3 x - 3 x \cdot 5}{5} - 6 \leq 0$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Factor $3 x$ out of $3 x - 3 x \cdot 5$ .

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Step 2.5.1.1.1

Factor $3 x$ out of $3 x$ .

$\frac{3 x (1) - 3 x \cdot 5}{5} - 6 \leq 0$

Step 2.5.1.1.2

Factor $3 x$ out of $- 3 x \cdot 5$ .

$\frac{3 x (1) + 3 x (- 1 \cdot 5)}{5} - 6 \leq 0$

Step 2.5.1.1.3

Factor $3 x$ out of $3 x (1) + 3 x (- 1 \cdot 5)$ .

$\frac{3 x (1 - 1 \cdot 5)}{5} - 6 \leq 0$

Step 2.5.1.2

Multiply $- 1$ by $5$ .

$\frac{3 x (1 - 5)}{5} - 6 \leq 0$

Step 2.5.1.3

Subtract $5$ from $1$ .

$\frac{3 x \cdot - 4}{5} - 6 \leq 0$

Step 2.5.1.4

Multiply $- 4$ by $3$ .

$\frac{- 12 x}{5} - 6 \leq 0$

Step 2.5.2

Move the negative in front of the fraction.

$- \frac{12 x}{5} - 6 \leq 0$

Step 3

Add $6$ to both sides of the inequality.

$- \frac{12 x}{5} \leq 6$

Step 4

Divide each term in $- \frac{12 x}{5} \leq 6$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{12 x}{5} \leq 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{12 x}{5}}{- 1} \geq \frac{6}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{12 x}{5}}{1} \geq \frac{6}{- 1}$

Step 4.2.2

Divide $\frac{12 x}{5}$ by $1$ .

$\frac{12 x}{5} \geq \frac{6}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $6$ by $- 1$ .

$\frac{12 x}{5} \geq - 6$

Step 5

Multiply both sides by $5$ .

$\frac{12 x}{5} \cdot 5 \geq - 6 \cdot 5$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $5$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{12 x}{5} \cdot 5 \geq - 6 \cdot 5$

Step 6.1.1.2

Rewrite the expression.

$12 x \geq - 6 \cdot 5$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- 6$ by $5$ .

$12 x \geq - 30$

Step 7

Divide each term in $12 x \geq - 30$ by $12$ and simplify.

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Step 7.1

Divide each term in $12 x \geq - 30$ by $12$ .

$\frac{12 x}{12} \geq \frac{- 30}{12}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $12$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{12 x}{12} \geq \frac{- 30}{12}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x \geq \frac{- 30}{12}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $- 30$ and $12$ .

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Step 7.3.1.1

Factor $6$ out of $- 30$ .

$x \geq \frac{6 (- 5)}{12}$

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Factor $6$ out of $12$ .

$x \geq \frac{6 \cdot - 5}{6 \cdot 2}$

Step 7.3.1.2.2

Cancel the common factor.

$x \geq \frac{6 \cdot - 5}{6 \cdot 2}$

Step 7.3.1.2.3

Rewrite the expression.

$x \geq \frac{- 5}{2}$

Step 7.3.2

Move the negative in front of the fraction.

$x \geq - \frac{5}{2}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x \geq - \frac{5}{2}$

Interval Notation:

$[- \frac{5}{2}, \infty)$