Algebra Examples

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

Step 1

Rewrite so $x$ is on the left side of the inequality.

$6 x - (- \frac{1}{2} x + 1) - 3 < - x$

Step 2

Simplify $6 x - (- \frac{1}{2} x + 1) - 3$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$6 x - (- \frac{x}{2} + 1) - 3 < - x$

Step 2.1.2

Apply the distributive property.

$6 x - - \frac{x}{2} - 1 \cdot 1 - 3 < - x$

Step 2.1.3

Multiply $- - \frac{x}{2}$ .

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Step 2.1.3.1

Multiply $- 1$ by $- 1$ .

$6 x + 1 \frac{x}{2} - 1 \cdot 1 - 3 < - x$

Step 2.1.3.2

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

$6 x + \frac{x}{2} - 1 \cdot 1 - 3 < - x$

Step 2.1.4

Multiply $- 1$ by $1$ .

$6 x + \frac{x}{2} - 1 - 3 < - x$

Step 2.2

To write $6 x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 x \cdot \frac{2}{2} + \frac{x}{2} - 1 - 3 < - x$

Step 2.3

Combine $6 x$ and $\frac{2}{2}$ .

$\frac{6 x \cdot 2}{2} + \frac{x}{2} - 1 - 3 < - x$

Step 2.4

Combine the numerators over the common denominator.

$\frac{6 x \cdot 2 + x}{2} - 1 - 3 < - x$

Step 2.5

Find the common denominator.

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Step 2.5.1

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

$\frac{6 x \cdot 2 + x}{2} + \frac{- 1}{1} - 3 < - x$

Step 2.5.2

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

$\frac{6 x \cdot 2 + x}{2} + \frac{- 1}{1} \cdot \frac{2}{2} - 3 < - x$

Step 2.5.3

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

$\frac{6 x \cdot 2 + x}{2} + \frac{- 1 \cdot 2}{2} - 3 < - x$

Step 2.5.4

Write $- 3$ as a fraction with denominator $1$ .

$\frac{6 x \cdot 2 + x}{2} + \frac{- 1 \cdot 2}{2} + \frac{- 3}{1} < - x$

Step 2.5.5

Multiply $\frac{- 3}{1}$ by $\frac{2}{2}$ .

$\frac{6 x \cdot 2 + x}{2} + \frac{- 1 \cdot 2}{2} + \frac{- 3}{1} \cdot \frac{2}{2} < - x$

Step 2.5.6

Multiply $\frac{- 3}{1}$ by $\frac{2}{2}$ .

$\frac{6 x \cdot 2 + x}{2} + \frac{- 1 \cdot 2}{2} + \frac{- 3 \cdot 2}{2} < - x$

Step 2.6

Combine the numerators over the common denominator.

$\frac{6 x \cdot 2 + x - 1 \cdot 2 - 3 \cdot 2}{2} < - x$

Step 2.7

Simplify each term.

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Step 2.7.1

Multiply $2$ by $6$ .

$\frac{12 x + x - 1 \cdot 2 - 3 \cdot 2}{2} < - x$

Step 2.7.2

Multiply $- 1$ by $2$ .

$\frac{12 x + x - 2 - 3 \cdot 2}{2} < - x$

Step 2.7.3

Multiply $- 3$ by $2$ .

$\frac{12 x + x - 2 - 6}{2} < - x$

Step 2.8

Simplify by adding terms.

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Step 2.8.1

Add $12 x$ and $x$ .

$\frac{13 x - 2 - 6}{2} < - x$

Step 2.8.2

Subtract $6$ from $- 2$ .

$\frac{13 x - 8}{2} < - x$

Step 3

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

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Step 3.1

Add $x$ to both sides of the inequality.

$\frac{13 x - 8}{2} + x < 0$

Step 3.2

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

$\frac{13 x - 8}{2} + x \cdot \frac{2}{2} < 0$

Step 3.3

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

$\frac{13 x - 8}{2} + \frac{x \cdot 2}{2} < 0$

Step 3.4

Combine the numerators over the common denominator.

$\frac{13 x - 8 + x \cdot 2}{2} < 0$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Move $2$ to the left of $x$ .

$\frac{13 x - 8 + 2 \cdot x}{2} < 0$

Step 3.5.2

Add $13 x$ and $2 x$ .

$\frac{15 x - 8}{2} < 0$

Step 4

Multiply both sides by $2$ .

$\frac{15 x - 8}{2} \cdot 2 < 0 \cdot 2$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{15 x - 8}{2} \cdot 2 < 0 \cdot 2$

Step 5.1.1.2

Rewrite the expression.

$15 x - 8 < 0 \cdot 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $0$ by $2$ .

$15 x - 8 < 0$

Step 6

Solve for $x$ .

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Step 6.1

Add $8$ to both sides of the inequality.

$15 x < 8$

Step 6.2

Divide each term in $15 x < 8$ by $15$ and simplify.

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Step 6.2.1

Divide each term in $15 x < 8$ by $15$ .

$\frac{15 x}{15} < \frac{8}{15}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $15$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{15 x}{15} < \frac{8}{15}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x < \frac{8}{15}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{8}{15}$

Interval Notation:

$(- \infty, \frac{8}{15})$