Algebra Examples

$- 2 + \frac{1}{2} x > 3 x - 9 (- \frac{1}{2} x - 1)$

Step 1

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

$3 x - 9 (- \frac{1}{2} x - 1) < - 2 + \frac{1}{2} x$

Step 2

Simplify $3 x - 9 (- \frac{1}{2} x - 1)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$3 x - 9 (- \frac{x}{2} - 1) < - 2 + \frac{1}{2} x$

Step 2.1.2

Apply the distributive property.

$3 x - 9 (- \frac{x}{2}) - 9 \cdot - 1 < - 2 + \frac{1}{2} x$

Step 2.1.3

Multiply $- 9 (- \frac{x}{2})$ .

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Step 2.1.3.1

Multiply $- 1$ by $- 9$ .

$3 x + 9 \frac{x}{2} - 9 \cdot - 1 < - 2 + \frac{1}{2} x$

Step 2.1.3.2

Combine $9$ and $\frac{x}{2}$ .

$3 x + \frac{9 x}{2} - 9 \cdot - 1 < - 2 + \frac{1}{2} x$

Step 2.1.4

Multiply $- 9$ by $- 1$ .

$3 x + \frac{9 x}{2} + 9 < - 2 + \frac{1}{2} x$

Step 2.2

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

$3 x \cdot \frac{2}{2} + \frac{9 x}{2} + 9 < - 2 + \frac{1}{2} x$

Step 2.3

Simplify terms.

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Step 2.3.1

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

$\frac{3 x \cdot 2}{2} + \frac{9 x}{2} + 9 < - 2 + \frac{1}{2} x$

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{3 x \cdot 2 + 9 x}{2} + 9 < - 2 + \frac{1}{2} x$

Step 2.4

Simplify the numerator.

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Step 2.4.1

Factor $3 x$ out of $3 x \cdot 2 + 9 x$ .

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Step 2.4.1.1

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

$\frac{3 x (2) + 9 x}{2} + 9 < - 2 + \frac{1}{2} x$

Step 2.4.1.2

Factor $3 x$ out of $9 x$ .

$\frac{3 x (2) + 3 x (3)}{2} + 9 < - 2 + \frac{1}{2} x$

Step 2.4.1.3

Factor $3 x$ out of $3 x (2) + 3 x (3)$ .

$\frac{3 x (2 + 3)}{2} + 9 < - 2 + \frac{1}{2} x$

Step 2.4.2

Add $2$ and $3$ .

$\frac{3 x \cdot 5}{2} + 9 < - 2 + \frac{1}{2} x$

Step 2.4.3

Multiply $5$ by $3$ .

$\frac{15 x}{2} + 9 < - 2 + \frac{1}{2} x$

Step 3

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

$\frac{15 x}{2} + 9 < - 2 + \frac{x}{2}$

Step 4

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

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Step 4.1

Subtract $\frac{x}{2}$ from both sides of the inequality.

$\frac{15 x}{2} + 9 - \frac{x}{2} < - 2$

Step 4.2

Combine the numerators over the common denominator.

$9 + \frac{15 x - x}{2} < - 2$

Step 4.3

Subtract $x$ from $15 x$ .

$9 + \frac{14 x}{2} < - 2$

Step 4.4

Cancel the common factor of $14$ and $2$ .

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Step 4.4.1

Factor $2$ out of $14 x$ .

$9 + \frac{2 (7 x)}{2} < - 2$

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Factor $2$ out of $2$ .

$9 + \frac{2 (7 x)}{2 (1)} < - 2$

Step 4.4.2.2

Cancel the common factor.

$9 + \frac{2 (7 x)}{2 \cdot 1} < - 2$

Step 4.4.2.3

Rewrite the expression.

$9 + \frac{7 x}{1} < - 2$

Step 4.4.2.4

Divide $7 x$ by $1$ .

$9 + 7 x < - 2$

Step 5

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

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Step 5.1

Subtract $9$ from both sides of the inequality.

$7 x < - 2 - 9$

Step 5.2

Subtract $9$ from $- 2$ .

$7 x < - 11$

Step 6

Divide each term in $7 x < - 11$ by $7$ and simplify.

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Step 6.1

Divide each term in $7 x < - 11$ by $7$ .

$\frac{7 x}{7} < \frac{- 11}{7}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $7$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{7 x}{7} < \frac{- 11}{7}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x < \frac{- 11}{7}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$x < - \frac{11}{7}$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x < - \frac{11}{7}$

Interval Notation:

$(- \infty, - \frac{11}{7})$