Algebra Examples

$- x \leq \frac{3}{2} + \frac{3}{2} (- 2 x - 10)$

Step 1

Simplify $\frac{3}{2} + \frac{3}{2} \cdot (- 2 x - 10)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$- x \leq \frac{3}{2} + \frac{3}{2} (- 2 x) + \frac{3}{2} \cdot - 10$

Step 1.1.2

Cancel the common factor of $2$ .

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Step 1.1.2.1

Factor $2$ out of $- 2 x$ .

$- x \leq \frac{3}{2} + \frac{3}{2} (2 (- x)) + \frac{3}{2} \cdot - 10$

Step 1.1.2.2

Cancel the common factor.

$- x \leq \frac{3}{2} + \frac{3}{2} (2 (- x)) + \frac{3}{2} \cdot - 10$

Step 1.1.2.3

Rewrite the expression.

$- x \leq \frac{3}{2} + 3 (- x) + \frac{3}{2} \cdot - 10$

Step 1.1.3

Multiply $- 1$ by $3$ .

$- x \leq \frac{3}{2} - 3 x + \frac{3}{2} \cdot - 10$

Step 1.1.4

Cancel the common factor of $2$ .

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Step 1.1.4.1

Factor $2$ out of $- 10$ .

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

Step 1.1.4.2

Cancel the common factor.

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

Step 1.1.4.3

Rewrite the expression.

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

Step 1.1.5

Multiply $3$ by $- 5$ .

$- x \leq \frac{3}{2} - 3 x - 15$

Step 1.2

To write $- 15$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- x \leq - 3 x + \frac{3}{2} - 15 \cdot \frac{2}{2}$

Step 1.3

Combine $- 15$ and $\frac{2}{2}$ .

$- x \leq - 3 x + \frac{3}{2} + \frac{- 15 \cdot 2}{2}$

Step 1.4

Combine the numerators over the common denominator.

$- x \leq - 3 x + \frac{3 - 15 \cdot 2}{2}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $- 15$ by $2$ .

$- x \leq - 3 x + \frac{3 - 30}{2}$

Step 1.5.2

Subtract $30$ from $3$ .

$- x \leq - 3 x + \frac{- 27}{2}$

Step 1.6

Move the negative in front of the fraction.

$- x \leq - 3 x - \frac{27}{2}$

Step 2

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

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Step 2.1

Add $3 x$ to both sides of the inequality.

$- x + 3 x \leq - \frac{27}{2}$

Step 2.2

Add $- x$ and $3 x$ .

$2 x \leq - \frac{27}{2}$

Step 3

Divide each term in $2 x \leq - \frac{27}{2}$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x \leq - \frac{27}{2}$ by $2$ .

$\frac{2 x}{2} \leq \frac{- \frac{27}{2}}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} \leq \frac{- \frac{27}{2}}{2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- \frac{27}{2}}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x \leq - \frac{27}{2} \cdot \frac{1}{2}$

Step 3.3.2

Multiply $- \frac{27}{2} \cdot \frac{1}{2}$ .

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Step 3.3.2.1

Multiply $\frac{1}{2}$ by $\frac{27}{2}$ .

$x \leq - \frac{27}{2 \cdot 2}$

Step 3.3.2.2

Multiply $2$ by $2$ .

$x \leq - \frac{27}{4}$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{27}{4}$

Interval Notation:

$(- \infty, - \frac{27}{4}]$