Algebra Examples

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

Step 1

Simplify $- (3 x + 1) - \frac{1}{2} x$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Multiply $3$ by $- 1$ .

$- 3 x - 1 \cdot 1 - \frac{1}{2} x > - \frac{3}{2} x - \frac{4}{5}$

Step 1.1.3

Multiply $- 1$ by $1$ .

$- 3 x - 1 - \frac{1}{2} x > - \frac{3}{2} x - \frac{4}{5}$

Step 1.1.4

Combine $x$ and $\frac{1}{2}$ .

$- 3 x - 1 - \frac{x}{2} > - \frac{3}{2} x - \frac{4}{5}$

Step 1.2

To write $- 3 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 3 x \cdot \frac{2}{2} - \frac{x}{2} - 1 > - \frac{3}{2} x - \frac{4}{5}$

Step 1.3

Simplify terms.

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Step 1.3.1

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

$\frac{- 3 x \cdot 2}{2} - \frac{x}{2} - 1 > - \frac{3}{2} x - \frac{4}{5}$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{- 3 x \cdot 2 - x}{2} - 1 > - \frac{3}{2} x - \frac{4}{5}$

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 $- 3 x \cdot 2 - x$ .

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Step 1.4.1.1.1

Factor $x$ out of $- 3 x \cdot 2$ .

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

Step 1.4.1.1.2

Factor $x$ out of $- x$ .

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

Step 1.4.1.1.3

Factor $x$ out of $x (- 3 \cdot 2) + x \cdot - 1$ .

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

Step 1.4.1.2

Multiply $- 3$ by $2$ .

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

Step 1.4.1.3

Subtract $1$ from $- 6$ .

$\frac{x \cdot - 7}{2} - 1 > - \frac{3}{2} x - \frac{4}{5}$

Step 1.4.2

Move $- 7$ to the left of $x$ .

$\frac{- 7 \cdot x}{2} - 1 > - \frac{3}{2} x - \frac{4}{5}$

Step 1.4.3

Move the negative in front of the fraction.

$- \frac{7 x}{2} - 1 > - \frac{3}{2} x - \frac{4}{5}$

Step 2

Simplify each term.

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Step 2.1

Combine $x$ and $\frac{3}{2}$ .

$- \frac{7 x}{2} - 1 > - \frac{x \cdot 3}{2} - \frac{4}{5}$

Step 2.2

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

$- \frac{7 x}{2} - 1 > - \frac{3 x}{2} - \frac{4}{5}$

Step 3

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

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Step 3.1

Add $\frac{3 x}{2}$ to both sides of the inequality.

$- \frac{7 x}{2} - 1 + \frac{3 x}{2} > - \frac{4}{5}$

Step 3.2

Combine the numerators over the common denominator.

$- 1 + \frac{- 7 x + 3 x}{2} > - \frac{4}{5}$

Step 3.3

Add $- 7 x$ and $3 x$ .

$- 1 + \frac{- 4 x}{2} > - \frac{4}{5}$

Step 3.4

Cancel the common factor of $- 4$ and $2$ .

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $2$ .

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

$- 1 + \frac{- 2 x}{1} > - \frac{4}{5}$

Step 3.4.2.4

Divide $- 2 x$ by $1$ .

$- 1 - 2 x > - \frac{4}{5}$

Step 4

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

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Step 4.1

Add $1$ to both sides of the inequality.

$- 2 x > - \frac{4}{5} + 1$

Step 4.2

Write $1$ as a fraction with a common denominator.

$- 2 x > - \frac{4}{5} + \frac{5}{5}$

Step 4.3

Combine the numerators over the common denominator.

$- 2 x > \frac{- 4 + 5}{5}$

Step 4.4

Add $- 4$ and $5$ .

$- 2 x > \frac{1}{5}$

Step 5

Divide each term in $- 2 x > \frac{1}{5}$ by $- 2$ and simplify.

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Step 5.1

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

$\frac{- 2 x}{- 2} < \frac{\frac{1}{5}}{- 2}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} < \frac{\frac{1}{5}}{- 2}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x < \frac{\frac{1}{5}}{- 2}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x < \frac{1}{5} \cdot \frac{1}{- 2}$

Step 5.3.2

Move the negative in front of the fraction.

$x < \frac{1}{5} (- \frac{1}{2})$

Step 5.3.3

Multiply $\frac{1}{5} (- \frac{1}{2})$ .

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Step 5.3.3.1

Multiply $\frac{1}{5}$ by $\frac{1}{2}$ .

$x < - \frac{1}{5 \cdot 2}$

Step 5.3.3.2

Multiply $5$ by $2$ .

$x < - \frac{1}{10}$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{1}{10}$

Interval Notation:

$(- \infty, - \frac{1}{10})$