Algebra Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Cancel the common factor of $2$ .

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Step 1.1.2.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

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

Step 1.1.2.2

Factor $2$ out of $- 4 x$ .

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

Step 1.1.2.3

Cancel the common factor.

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

Step 1.1.2.4

Rewrite the expression.

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

Step 1.1.3

Multiply $- 2$ by $- 3$ .

$- 2 x + 6 x - \frac{3}{2} \cdot 5 \leq - \frac{3}{2}$

Step 1.1.4

Multiply $- \frac{3}{2} \cdot 5$ .

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Step 1.1.4.1

Multiply $5$ by $- 1$ .

$- 2 x + 6 x - 5 (\frac{3}{2}) \leq - \frac{3}{2}$

Step 1.1.4.2

Combine $- 5$ and $\frac{3}{2}$ .

$- 2 x + 6 x + \frac{- 5 \cdot 3}{2} \leq - \frac{3}{2}$

Step 1.1.4.3

Multiply $- 5$ by $3$ .

$- 2 x + 6 x + \frac{- 15}{2} \leq - \frac{3}{2}$

Step 1.1.5

Move the negative in front of the fraction.

$- 2 x + 6 x - \frac{15}{2} \leq - \frac{3}{2}$

Step 1.2

Add $- 2 x$ and $6 x$ .

$4 x - \frac{15}{2} \leq - \frac{3}{2}$

Step 2

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

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Step 2.1

Add $\frac{15}{2}$ to both sides of the inequality.

$4 x \leq - \frac{3}{2} + \frac{15}{2}$

Step 2.2

Combine the numerators over the common denominator.

$4 x \leq \frac{- 3 + 15}{2}$

Step 2.3

Add $- 3$ and $15$ .

$4 x \leq \frac{12}{2}$

Step 2.4

Divide $12$ by $2$ .

$4 x \leq 6$

Step 3

Divide each term in $4 x \leq 6$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 x \leq 6$ by $4$ .

$\frac{4 x}{4} \leq \frac{6}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} \leq \frac{6}{4}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{6}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $6$ and $4$ .

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Step 3.3.1.1

Factor $2$ out of $6$ .

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

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $4$ .

$x \leq \frac{2 \cdot 3}{2 \cdot 2}$

Step 3.3.1.2.2

Cancel the common factor.

$x \leq \frac{2 \cdot 3}{2 \cdot 2}$

Step 3.3.1.2.3

Rewrite the expression.

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

Step 4

The result can be shown in multiple forms.

Inequality Form:

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

Interval Notation:

$(- \infty, \frac{3}{2}]$