Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

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

Step 2.1.2

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

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Step 2.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.1.2.2

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

$\frac{1}{2} x - \frac{1}{2} \cdot 2 + x \geq x + 5$

Step 2.1.2.3

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

$\frac{x}{2} - \frac{1}{2} \cdot 2 + x \geq x + 5$

Step 2.1.3

Cancel the common factor of $2$ .

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Step 2.1.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$\frac{x}{2} + \frac{- 1}{2} \cdot 2 + x \geq x + 5$

Step 2.1.3.2

Cancel the common factor.

$\frac{x}{2} + \frac{- 1}{2} \cdot 2 + x \geq x + 5$

Step 2.1.3.3

Rewrite the expression.

$\frac{x}{2} - 1 + x \geq x + 5$

Step 2.2

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

$\frac{x}{2} + x \cdot \frac{2}{2} - 1 \geq x + 5$

Step 2.3

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

$\frac{x}{2} + \frac{x \cdot 2}{2} - 1 \geq x + 5$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x + x \cdot 2}{2} - 1 \geq x + 5$

Step 2.5

Add $x$ and $x \cdot 2$ .

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Step 2.5.1

Reorder $x$ and $2$ .

$\frac{x + 2 \cdot x}{2} - 1 \geq x + 5$

Step 2.5.2

Add $x$ and $2 \cdot x$ .

$\frac{3 x}{2} - 1 \geq x + 5$

Step 3

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

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Step 3.1

Subtract $x$ from both sides of the inequality.

$\frac{3 x}{2} - 1 - x \geq 5$

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify each term.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

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

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Step 3.5.1.1.1

Factor $x$ out of $3 x$ .

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

Step 3.5.1.1.2

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

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

Step 3.5.1.1.3

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

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

Step 3.5.1.2

Multiply $- 1$ by $2$ .

$\frac{x (3 - 2)}{2} - 1 \geq 5$

Step 3.5.1.3

Subtract $2$ from $3$ .

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

Step 3.5.2

Multiply $x$ by $1$ .

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

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.

$\frac{x}{2} \geq 5 + 1$

Step 4.2

Add $5$ and $1$ .

$\frac{x}{2} \geq 6$

Step 5

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \geq 6 \cdot 2$

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.

$\frac{x}{2} \cdot 2 \geq 6 \cdot 2$

Step 6.1.1.2

Rewrite the expression.

$x \geq 6 \cdot 2$

Step 6.2

Simplify the right side.

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Step 6.2.1

Multiply $6$ by $2$ .

$x \geq 12$

Step 7

The result can be shown in multiple forms.

Inequality Form:

$x \geq 12$

Interval Notation:

$[12, \infty)$