Algebra Examples

$x - 2 \geq 10 + 9 (\frac{1}{2} x - 1)$

Step 1

Simplify $10 + 9 (\frac{1}{2} x - 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$x - 2 \geq 10 + 9 (\frac{x}{2} - 1)$

Step 1.1.2

Apply the distributive property.

$x - 2 \geq 10 + 9 \frac{x}{2} + 9 \cdot - 1$

Step 1.1.3

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

$x - 2 \geq 10 + \frac{9 x}{2} + 9 \cdot - 1$

Step 1.1.4

Multiply $9$ by $- 1$ .

$x - 2 \geq 10 + \frac{9 x}{2} - 9$

Step 1.2

Subtract $9$ from $10$ .

$x - 2 \geq \frac{9 x}{2} + 1$

Step 2

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

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Step 2.1

Subtract $\frac{9 x}{2}$ from both sides of the inequality.

$x - 2 - \frac{9 x}{2} \geq 1$

Step 2.2

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

$x \cdot \frac{2}{2} - \frac{9 x}{2} - 2 \geq 1$

Step 2.3

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

$\frac{x \cdot 2}{2} - \frac{9 x}{2} - 2 \geq 1$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - 9 x}{2} - 2 \geq 1$

Step 2.5

Subtract $9 x$ from $x \cdot 2$ .

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Step 2.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - 9 x}{2} - 2 \geq 1$

Step 2.5.2

Subtract $9 x$ from $2 \cdot x$ .

$\frac{- 7 \cdot x}{2} - 2 \geq 1$

Step 2.6

Move the negative in front of the fraction.

$- \frac{7 x}{2} - 2 \geq 1$

Step 3

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

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Step 3.1

Add $2$ to both sides of the inequality.

$- \frac{7 x}{2} \geq 1 + 2$

Step 3.2

Add $1$ and $2$ .

$- \frac{7 x}{2} \geq 3$

Step 4

Divide each term in $- \frac{7 x}{2} \geq 3$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- \frac{7 x}{2} \geq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{7 x}{2}}{- 1} \leq \frac{3}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{7 x}{2}}{1} \leq \frac{3}{- 1}$

Step 4.2.2

Divide $\frac{7 x}{2}$ by $1$ .

$\frac{7 x}{2} \leq \frac{3}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $3$ by $- 1$ .

$\frac{7 x}{2} \leq - 3$

Step 5

Multiply both sides by $2$ .

$\frac{7 x}{2} \cdot 2 \leq - 3 \cdot 2$

Step 6

Simplify.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1

Cancel the common factor.

$\frac{7 x}{2} \cdot 2 \leq - 3 \cdot 2$

Step 6.1.1.2

Rewrite the expression.

$7 x \leq - 3 \cdot 2$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $- 3$ by $2$ .

$7 x \leq - 6$

Step 7

Divide each term in $7 x \leq - 6$ by $7$ and simplify.

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Step 7.1

Divide each term in $7 x \leq - 6$ by $7$ .

$\frac{7 x}{7} \leq \frac{- 6}{7}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $7$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{7 x}{7} \leq \frac{- 6}{7}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x \leq \frac{- 6}{7}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Move the negative in front of the fraction.

$x \leq - \frac{6}{7}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x \leq - \frac{6}{7}$

Interval Notation:

$(- \infty, - \frac{6}{7}]$