Algebra Examples

$\frac{x + 1}{3} - \frac{2 x - 4}{6} \leq - \frac{x}{2}$

Step 1

Multiply both sides by $2$ .

$(\frac{x + 1}{3} - \frac{2 x - 4}{6}) \cdot 2 \leq - \frac{x}{2} \cdot 2$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(\frac{x + 1}{3} - \frac{2 x - 4}{6}) \cdot 2$ .

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Step 2.1.1.1

Simplify terms.

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Step 2.1.1.1.1

Cancel the common factor of $2 x - 4$ and $6$ .

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Step 2.1.1.1.1.1

Factor $2$ out of $2 x$ .

$(\frac{x + 1}{3} - \frac{2 (x) - 4}{6}) \cdot 2 \leq - \frac{x}{2} \cdot 2$

Step 2.1.1.1.1.2

Factor $2$ out of $- 4$ .

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

Step 2.1.1.1.1.3

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

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

Step 2.1.1.1.1.4

Cancel the common factors.

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Step 2.1.1.1.1.4.1

Factor $2$ out of $6$ .

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

Step 2.1.1.1.1.4.2

Cancel the common factor.

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

Step 2.1.1.1.1.4.3

Rewrite the expression.

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

Step 2.1.1.1.2

Combine the numerators over the common denominator.

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

Step 2.1.1.2

Simplify each term.

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Step 2.1.1.2.1

Apply the distributive property.

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

Step 2.1.1.2.2

Multiply $- 1$ by $- 2$ .

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

Step 2.1.1.3

Simplify terms.

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Step 2.1.1.3.1

Combine the opposite terms in $x + 1 - x + 2$ .

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Step 2.1.1.3.1.1

Subtract $x$ from $x$ .

$\frac{0 + 1 + 2}{3} \cdot 2 \leq - \frac{x}{2} \cdot 2$

Step 2.1.1.3.1.2

Add $0$ and $1$ .

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

Step 2.1.1.3.2

Add $1$ and $2$ .

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

Step 2.1.1.3.3

Cancel the common factor of $3$ .

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Step 2.1.1.3.3.1

Cancel the common factor.

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

Step 2.1.1.3.3.2

Rewrite the expression.

$1 \cdot 2 \leq - \frac{x}{2} \cdot 2$

Step 2.1.1.3.4

Multiply $2$ by $1$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factor.

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

Step 2.2.1.3

Rewrite the expression.

$2 \leq - x$

Step 3

Solve for $x$ .

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Step 3.1

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

$- x \geq 2$

Step 3.2

Divide each term in $- x \geq 2$ by $- 1$ and simplify.

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2

Divide $x$ by $1$ .

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $2$ by $- 1$ .

$x \leq - 2$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$x \leq - 2$

Interval Notation:

$(- \infty, - 2]$